TIGHT BINGING BOOK TEXT IS FLY < ! [<OU_164519 | CO MONOGRAFIE MATEMATYCZNE KOMITET REDAKCYJNY S. BANACH B. KNASTER K. KURATOWSKl S. MAZURKIEWICZ W. SIERPltiSKI H. STEINHAUS A. ZYGMUND TOM VII Theory of the Integral BY DR. STANISLAW SAKS Second Revised Edition ENGLISH TRANSLATION BY L. C. YOUNG, M.A." WITH TWO ADDITIONAL NOTES BY PROF. DR. STEFAN BANACH HAFNER PUBLISHING COMPANY. NEW YORK Printed in United Stale* of A Photolithographed by The Murray Printing Company Wakefield. Massachusetts PEEFACE. This edition differs from the first l ) by the new arrangement of the contents of several chapters, some of which have been completed by more recent results, and by the suppression of a number of errors, obligingly pointed out by Mr. V. Jarnik, which formed the object of the two pages of Errata in the first edition. It is probable that fresh errors have slipped in owing to modifications of the text, but the reader would certainly find many more, if the author had not received the valuable help of Messrs J. Todd, A. J. Ward and A. Zygmund in reading the proofs. Also, Mr. L. C. Young has greatly exceeded his rdle of translator in his collaboration with the author. To all these I express my warmest thanks. This volume contains two Notes by S. Banach. The first of them, on Haar's measure, is the translation (with a few slight modifications) of the note already contained in the French edition of this book. The second, which concerns the integration in abstract spaces, is published here for the first time and com- pletes the considerations of Chapter I. The numbers given in the bibliographical references relate to the list of cited works which will be found at the end of the book. The asterisks preceding certain titles indicate the parts of the book which may be omitted on first reading. 8. Saks. Warszawa-Zoliborz, July, 1937. l ) 8. Saks, Theorie de I' Integrate, Monografie Matematyczne, Volume II, Warszawa 1933. FEOM THE PEEFACE TO THE FIEST EDITION. The modern theory of real functions became distinct from clas- sical analysis in the second half of the 19-th century, as a result of researches, unsystematic at first, which dealt with the foundations of the Differential Calculus or which concerned the discovery of functions whose properties appeared to be very strange and un- expected. The distrust with which this new field of investigation was regarded is typified by the attitude of H. Poincar6 who wrote :" Autre- fois quand on inventait une fonction nouvelle, tfetait en vue de quelque but pratique; aujourd'hui on les invente tout exprds pour mettre en defaut les raisonnements de nos pdres et on n'en tirera jamais que cela". This view was by no means isolated. Ch. Hermite, in a letter to T. J. Stieltjes, expressed himself in even stronger terms: "Je me detourne avec effroi et horreur de cette plaie lamentable des fonctions qui riont pas de d&rivfos". Eesearches dealing with non-analytic functions and with functions violating laws which one hoped were uni- versal, were regarded almost as the propagation of anarchy and chaos where past generations had sought order and harmony. Even the first attempts to establish a positive theory were rather sceptically received: it was feared that an excessively pedantic exactitude in formulating hypotheses would spoil the elegance of classical methods, and that discussions of details would end by obscuring the main ideas of analysis. It is true that the first researches hardly went beyond the traditional, formal apparatus, fixed by Cauchy and Riemann, which was difficult to adapt to the requirements of the new problems. Nevertheless, these researches succeeded in opening the way to applications of the Theory of Sets to Analysis, and to quote H. Lebesgue's inaugural lecture at the College de France "the great authority of Camille Jordan gave to the new school a valuable encouragement which amply compensated the few reproofs it had to suffer". E. Baire, E. Borel, H. Lebesgue these are the names which represent the Theory of. Real Functions, not merely as an object of re- searches, but also as a method, names which at the same time recall the leading ideas of the theory. The names of Baire and Borel will be always associated with the method of classification of functions and sets in a transfinite hierarchy by means of certain simple operations to which they are subjected. Excellent accounts of this subject are to be found in the treatises: Ch. J. de la Valise Poussin, Fonctions ft ensemble, Integrate de Lebesgue , Classes de Baire, 1916, F. Hausdorff, Mengenlehre, 1927, H. Hahn, Theorie der reellen Funktionen, 1933 (recent edition), C. Kuratowski, third volume of the present collection, and finally in the book of W. Sier- p in ski, Topologja ogolna (in Polish), and its English translation, Generbl Topology, to be published in 1934 by the Toronto Uni- versity Press. The other line of researches, which arises directly from the study of the foundations of the Integral Calculus, is still more in- timately connected with the great trains of thought of Analysis in the last century. On several occasions attempts were made to gen- eralize the old process of integration of Cauchy-Eiemann, but it was Lebesgue who first made real progress in this matter. At the same time, Lebesgue's merit is not only to have created a new and more general notion of integral, nor even to have established its intimate connection with the theory of measure: the value of his work consists primarily in his theory of derivation which is parallel to that of integration. This enabled his discovery to find many applications in the most widely different branches of Analysis and, from the point of view of method, made it possible to reunite the two fundamental conceptions of integral, namely that of definite integral and that of primitive, which appeared to be forever separated as soon as integration went outside the domain of continuous functions. The theory of Lebesgue constitutes the subject of the present volume. While distinguishing it from that of Baire, we have no wish to erect an artificial barrier between two streams of thought which naturally intermingle. On the contrary, we shall have frequent occasion, particularly in the last chapters of this book, to show expli- citly how Lebesgue's theory comes to be bound up not only with the results, but also with the very methods, of the theory of Baire. Is not the idea of Denjoy integration at bottom merely a striking adaptation of the idea which guided Baire? Where Baire, by repeated application of passage to the limit, widened the class of functions, Denjoy constructed a transfinite hierarchy of methods of integration starting with that of Lebesgue and whose successive stages are connected by two operations: one corresponding exactly to the generalized integral of Cauchy and the other to the generalized inte- gral of Harnack- Jordan. VI Now that the Theory of Beal Functions, while losing perhaps a little of the charm of its first youth, has ceased to be a "new" science, it seems superfluous to discuss its importance. It is known that the theory has brought to light regularity and harmony, unhoped for by the older methods, concerning, for instance, the existence of a limit, a derivative, or a tangent. It is enough to mention the theorems, now classical, on the behaviour of a power series on, or near, the boundary of its circle of convergence. Also, many branches of analysis, to cite only Harmonic Analysis, Integral Equa- tions, Functional Operations, have lost none of their elegance where they have been inspired by methods of the Theory of Real Functions. On the contrary, we have learnt to admire in the arguments not only cleverness of calculation, but also the generality which, by an apparent abstraction, often enables us to grasp the real nature of the problem. The object of the preceding remarks has been to indicate the place occupied by the subject of this volume in the Theory of Beal Functions 1 ). Let us now say a few words about the structure of the book. It embodies the greater part of a course of lectures delivered by the author at the University of Warsaw (and published in Polish in a separate book 2 )), which has been modified aud completed by several chapters. The reader need only be acquainted with a few elementary principles of the Theory of Sets, which are to t>e found in most courses of lectures on elementary analysis. Actually a sum- mary of the elements of the theory of sets of points is given in one of the opening paragraphs. Several pages of the book are inspired by suggestions and methods which I owe to the excellent university lectures of my teacher, ' W. Sierpiriski, the influence of whose ideas has often guided my personal researches. Finally, I wish to express my warmest thanks to all those who have kindly assisted me in my task, partic- ularly to my friend A. Zygmund, who undertook to read the manu- script. I thank also Messrs 0. Kuratowski and H. Steinhaus for their kind remarks aiul bibliographical indications. S. Saks. Warszawa, May, 1933. 1 ) In this preface, I made no attempt to write a history of the early Lays of the theory, and still less to settle questions of priority of discovery, lut, since an English Edition of this hook is appearing now, 1 think I ought to lention the name of W. H. Young, whose work on the theory of integration tarted at the same period as that of Lehesgue. 2 ) Zarys teorji calki, Warszawa 1930, Wydawnictwo Kasy im. Mianow- :iego, Instytutu Popierania Nauki. CHAPTER I. The integral in an abstract space. # 1. Introduction. Apart from functions having as argument a variable number, or system of n numbers (point in ^-dimensional space), we shall discuss in this book functions for which the inde- pendent variable is a set of points. Functions of this kind have occurred already in classical Analysis, in several important particular cases. But they only began to be studied in their full generality during the growth of the Theory of Sets, and in close relation to the parts of Analysis directly based on that theory. If we are, for instance, given a function f(x) integrable on every interval, then by associating with each interval / the value of the integral of f(x) over /, we obtain a function b\I) that is a function of an interval. Similarly, by taking multiple integrals of functions 1(' x n #'2? > *n) f n variables, we are led to consider functions of more general sets lying in spaces of several dimensions, the argu- ment I of our function F(l) being no\\ replaced by any set for which the integral of our given function /(,r t , ,r 2 , ..., x n ) is defined. We dwell on these examples in order to emphasize the natural connection between the notion of integral (in any sense) and that of function of a set. Nt^edless to say, there are many other examples of functions of a set. Thus in elementary geometry, we have for instance, the length of a segment or the. area of a polygon. The class of values of the argument of these two functions (the length and the, area) is in the first case, the class of segments and in the second, that of polygons. The problem of extending these classes gave birth to the general theories of measure, in which the notions of length, area, and volume, defined in elementary geometry for a restricted number of figures, are now extended to sets of points 2 CHAPTER I. The integral in an abstract space. of much greater diversity. It is, nevertheless, remarkable that these researches arose far less from problems of Geometry than from their connection with problems of Analysis, above all with the tendency to generalize, and to render more precise, the notion of definite integral. This connection has occasionally found expression even in the terminology. Thus duBois-Reymond called integrable the sets that to-day are said to be of measure zero in the Jordan sense. The theories of measure have, in the course of their development, been modified in accordance with the changing requirements of the Theory of Functions. In our account, the most important part will be played by the theory of H. Lebesgue. Lebesgue's theory of measure has made it possible to dis- tinguish in Euclidean spaces a vast class of sets, called measurable, in which measure has the property of complete additivity by this we mean that the measure of the sum of a sequence, even infinite, of measurable sets, no two of which have points in common, is equal to the sum of the measures of these sets. The importance of this class of sets is due to the fact that it includes, in particular, (with their classical measures), all the sets of points occurring in problems of classical Analysis, and further, that the fundamental operations applied to measurable sets lead always to measurable sets. It is nevertheless to be observed that the ground was prepared for Lebesgue's theory of measure by earlier theories associated with the names of Cantor, Stolz, Harnack, du Bois-Eeymond, Peano, Jordan, Borel, and others. These earlier theories have, however, to-day little more than historical value. They, too, were suitable instruments for studying and generalizing the notion of integral understood in the classical sense of Riemann, but their results in this direction have been largely artificial and accidental. It is only Lebesgue's theory of measure that makes a decisive step in the development of the notion of integral. This is the more re- markable in that the definition of Lebesgue apparently requires only a very small modification of a formal kind in the definition of integral due to Riemann. To fix the ideas, let us consider a bounded function /(#, y) of two variables, or what comes to the same thing, a bounded func- tion of a variable point defined on a square AT . In order to deter- mine its Riemann integral, or more precisely, its lower Riemann- Darboux integral over K^ we proceed as follows. We divide the t 1] Introduction. 3 square K Q into an arbitrary finite number of non-overlapping rectangles R^ /? 2 , ..., R nj and we form the sum (1.1) J>-m(ie,) where v, denotes the lower bound of the function / on /Z/, and m(R,) denotes the area of R,. The upper bound of all sums of this form is, by definition, the lower Riemann-Darboux integral of the function / over K . We define similarly the upper integral of / over K Q . If these two extreme integrals are equal, their common value is called the definite Riemann integral of the function / over jfiC , and the function / is said to be integrable in the Riemann sense over K Q . The extension of measure to all sets measurable in the Lebesgue sense, has rendered necessary a modification of the process of Bie- maiin-Darboux, it being natural to consider sums of the form (1.1) for which |/2/5/--i.2.i.../ is a subdivision of the square K Q into a finite number of arbitrary measurable sets, not necessarily either rectangles or elementary geometrical figures. Accordingly, m(Rj) is to be un- derstood to mean the measure of R lm The / retain their former meaning, i. e. represent the lower bounds of / on the corresponding gets Kj. We might call the upper bound of the sums (I. I) interpreted in this way the loiper Lebesgue integral of the function / over K . But actually, this process is of practical importance only for a class of functions, called measurable, and for these the number obtained as the upper bound of the sums (1.1) is called simply the definite Lebesgue integral of / over / . What is important, is that the func- tions which are measurable in the sense of Lebesgue, and whoso definition is closely related to that of the measurable sets, form a very general class. This class includes, in particular, all the func- tions integrable in the Riemann sense. Apart from this, tho method of Lebesgue is not only more general, but even, from a certain point of- view, simpler than that of Riemann-Darboux. For, it dispenses with the simultaneous introduction of two extreme integrals, tho lowor and t.ho upper. Thanks to this, Lebesgue's method loads itself to an immediate extension to unbounded functions, at any rate to certain rlassos of tho latter, for instance, to all moasurablo functions of constant sign (of. below 10). Finally, tho Lebesgue integral readers it por- missiblo to integrate torni by tonn soqueaeos and series of fuactioas in cortaia general rasas whore passages to tho limit nailer the ia- 4 CHAPTER I. The integral in an abstract space. tegral sign were not allowed by the earlier methods of integration. The reason for this is to be found in the complete additivity of Le- besgue measure. The fundamental theorems ofLebesgue (cf. below 12) stating the precise circumstances under which term by term integration is permissible, are justly regarded by Ch. J. de la Valise Poussin [I, p. 44] as one of the finest results of the theory. Lebesgue's theory of measure has, in its turn, led naturally to further important generalizations. Instead of starting with area, or volume, of figures, we may imagine a mass distributed in the Euclidean space under consideration, and associate with each set as its measure, (its "weight" according to Ch. J. de la Valise Poussin [I, Chap.VI; 1]), the amount of mass distributed on the set. This, again, leads to a generalization of the integral, parallel to Lebesgue's, known as the Lebesgue-Stieltjes integral. In order to present a unified account of the latter, we shall consider in this chapter an additive class of measurable sets given a priori in an arbitrary abstract space. We shall suppose further, that in this class, a completely additive measure is determined for the mea- surable sets. These hypotheses determine completely a corresponding method of integration in the Lebesgue sense. All the essential prop- erties of the ordinary Lebesgue integral, except at most those im- plying the process of derivation, remain valid for this abstract integral. From this point of view, in a more or less general form, the Lebesgue integral has been studied by a number of authors, among whom we may mention J. Radon [1], P. J. Daniell [2], O. Nikodym [2] and B. Jessen [1]. For further generalizations (of a somewhat different kind) see also A. Kolmogoroff [1], S. Bochner [1], G. Fichtenholz and L. Kantorovitch [1], and M. Gowu- rin [1]. $ 2. Terminology and notation. Given two sets A and B, we write A B when the set A is a subset of the set , i. e. when every element of A is an element of B. When we have both A C B and BA, i. e. when the sets A and B consist of the same ele- ments, we write A = B. Again, a eA means that a is an element of the set A (belongs to A). By the empty set, we mean the set without any element; we denote it by 0. A set A is enumerable if there existft an infinite sequence of distinct elements o 1? a 2 , .,., a,,, ... consisting of all the elements of the set A. [ 2] Terminology and notation. 5 Given a class $1 of sets, we call sum of the sets belonging to this class, the set of all the objects each of which is an element of at least one set belonging to the class 21. We call product, or common pari 9 of the sets belonging to the class % the set of all the objects that belong at the same time to all the sets of this class. We call difference of two sets A and B, and we denote by A B, the set of all the objects that belong to A without belonging to B. Given a sequence of sets [A,,] a finite sequence A^A^ ..., A nj or an infinite sequence A } ,A 2 , ..., A n , ... we denote the sum by 11 2 A,, by Aij or by A 1 + A 2 + ... + A n , in the finite case, and by / / i oo 2 A I9 by A,-, or by A l + A^+ ...+ A tl + ... in the infinite case. / /=rl Similarly, merely replacing the sign 2 by 77, we have the expres- sion for the product of a sequence of sets. If the sequence {A,,} is infinite, we call upper limit of this sequence, the set of all the ele- ments a such that a*A n holds for an infinity of values of the index n. The set of all the elements a belonging to all the sets A n from some n (in general depending on a) onwards, we call lower limit of the se- quence (A n \. The upper and lower limits of the sequence (A,,),, i, 2 ,..., we denote by lim sup ^4,, and lim inf A H respectively. We have /i ;i oo oo oo oo (2.1) lim inf A,, = 2 ft A-.C II 2 A,, = lim sup A,,. n ft =1 n^~k h\ n- k n If lim sup A n = lim inf A n * the sequence {A lt } is said to be convergent; n n its upper and lower limits are then called simply limit and denoted by lim ^4,,. // If, for a sequence {A,,} of sets, we have A,,C^/*+i> for each n, the sequence {A n } is said to be ascending, or non-decreasing ; if, for each ft, we have A n +\d.A nj the sequence [A,,] is said to be descending or non-increasing. Ascending and descending sequences are called monotone. We see directly that every monotone sequence is con- vergent, and that we have lim A n = 2A n for every ascending se- // // quence {A,,}, and lim A n = 77^4 for every descending sequence {A n }. n n Finally, given a class ( of sets, we shall often call the sets belonging to (, for short, sets ((). The class of the sets which are the sums of sequences of sets (() will be denoted by da. The class of the sets which are the products of such sequences will be denoted by (,> (see P. Hausdorff [II, p. 83]). 6 CHAPTER I. The integral in an abstract space. $ 3. Abstract space X. In the rest of this chapter, a set A" will be fixed and called apace. The elements of -V will be called points. If A is any set contained in A" the set A' A will be called com- plement of A with respect to A"; the expression "with respect to A" will, however, generally be omitted, since sets outside the space -T will not be considered. The complement of a set A will be denoted by CA. We evidently have, for every pair of sets A and B, (3.1) A B = A-CB, and for every sequence {JKV/ of sets 77 Z n = C2CX,, , 2 Z H = C 77 CZ n , (3.2) lim sup Z,t = C lim inf CZ n , lim inf X n = C lim sup CZ H . n n n n In the space A" we shall consider functions of a set, and functions of a point. The values of these functions will always be real numbers, finite or infinite. A function will be called finite, when it assumes only finite values. To avoid misunderstanding, let us agree that when infinite func- tions are subjected to the elementary operations of addition, subtrac- tion etc., we make the following conventions: a+( oo)=( o)+a= 00 for a+Ifoo; ( +00 )+( oo) = ( oo)+( + oo)==(cx>)_(oo)=0; -( 00 )=( 00 ) a=oo and a- ( oo)=(:t 00 ) . a= : f : oo, according as a>0 or a<0; 0-(oo)=(oo) -0=0; a/(oo)=0; a/0=+oo. We call characteristic junction C K (X) of a set E, the function (of a point) equal to 1 at all points of the set, and to everywhere else. The following theorem is obvious: (3.3) // E=2E,,, and J?/..&*=0 whenever i^k, then G K (X)=^C K (x). it ' n ' n If {E n } is (i monotone sequence of sets, the sequence of their characteristic functions is also monotone, non-deo easing or non-in- creasing according as the sequence {E n } is ascending or descending. If {E' lt } is any sequence of sets, A and B denoting its upper and lower limits respectively, we hare C A (a?)=lim sup C K (x), and c B (x) = lim inf c (x) ; n '" n ' 90 that, in order that a sequence of sets {E n } converge to a set E, it is necessary and sufficient that the sequence of their character" ittic function* {c^(a)} converge to the function (C K (X)}. [ 4] Additive classes of sets. 7 A function assuming only a finite number of different values on a set E is called a simple junction on E. If t> lf v a ,..., v n are all the distinct values of a simple function f(x) on a set A\ the function /(a?) may on E be written in the form and J5,-i>=0 /or The function / given by this formula over the set E will be denoted by fa, jE^a, .S 2 ; ...; t?,,, }. The notion of characteristic function is due to Oh. J. de ia Vallee Poussin [1] and [I, p. 7]. $ 4. Additive classes of sets. A class 9E of sets in the space A will be called additive if (i) the empty set belongs to 9, (ii) when a set X belongs to 9 so does its complement CX. and (iii) the sum of a sequence {X,,} of sets selected from the class 2E, belongs also to the class 9E. The classes of sets, additive according to this definition, are sometimes termed completely additive. We get the definition of a class of sets additive in the weak sense if we replace the condition (iii) of the preceding definition by the following: (iii-bis) the snm of two seta belonging to $ also belongs to $. The sets of an additive class 9E will be called sets measurable (9E), or, in accordance with the definition given in 2 (p. 5), simply sets (9E). We see at once that, on account of the conditions (i) and (ii), the space A', as complement of the empty set, belongs to every additive class of sets. Making use of the relations (2.1), (3.1), and (3.2), we obtain immediately the following: (4.1) Theorem. If 9 is an additive class of setSj the sum, the product, and the two limits, upper and lower, of every sequence of sete measurable (9E), and the difference of two sets measurable (9E), are also measurable (9E). In later chapters we shall consider certain additive classes of sets that present themselves naturally to us, in connection with the theory of measure, in metrical or in Euclidean spaces. Tn the abstract space A r , about which we have made practically no hypo- thesis, we can only mention a few trivial examples of additive classes of sets, such as the class of all sets in A , or the class of all finite or enumerable sets and their complements. Let us still mention one further general theorem: 8 CHAPTER I. The integral in an abstract space. (4.2) Theorem. Given any class 3R of sets in X, there exists always a smallest additive class of sets containing 3R, i. e. an additive class 91 3^ contained in every other additive class that contains 2R. For let 91 be the product of all the additive classes that con- tain 3)1. Such classes evidently exist, one such class being the class of all sets in X. We see at once that the class 9t thus defined has the required properties. $ 5, Additive functions of a set. In the rest of this chapter we suppose that a definite additive class 2E of sets is fixed in the space X. In accordance with this hypothesis, we may often omit the symbol SE in our statements, without causing any ambiguity. A function of a set, ( / ; (-), will be called additive function of a set (9E) on a set E, if (i) E is a set (3E), (ii) the function <P(X) is defined and finite for each set X(^E measurable (SE), and if (iii) V(2X n ) = 2<D(X n ) for every sequence {X n } of sets (SE) con- // 11 tained in E and such that Jf/ X h whenever i ^Jc. For simplicity, we shall speak of an "additive function" instead of an "additive function of a set (SE)" whenever there is no mistaking the meaning. An additive function of a set (9E) will be called monotone on E if its values for the subsets (SE) of E are of constant sign. A non- negative function <P(SE) additive and monotone, will also be termed non-decreasing, on account of the fact that, for each pair of sets A and B measurable (SE), the inequality ACS implies $(B) = = <P(A) + <1>(B A) ^ $(A). For the same reason, non-positive monotone functions will be termed non-increasing. (5.1) Theorem. If <P(x) is an additive function on a set E, then (5.2) for every monotone sequence {X n } of sets (SE) contained in E. If is a non-negative monotone function, then (5.3) <P(liminf J^Xliminf $(X n ) and n n for every sequence {X,,} of sets (9E) in E. [{ 5] Additive functions of a set. 9 Proof. Let {JTn5-i,2,... be a sequence of sets (9) contained in IS. If {X n } is an ascending monotone sequence, then lim X n = j X n = X r + jf (XH-I X n ), n n 1 n=^l and consequently, #(JO being an additive function on E, 4>(KmX tl ) = 0(2^) + f 0(^+1 X,,) - l , J*)] - lim If {X,,} is a descending sequence, the sequence {E X,,} is ascending, and, by the result already proved, (f(E) 0(limX fl )==0[lim(J0 X fl )] = lim $(EX n )=0(E)1i from which (5.2) follows at once. Finally, if {X n } is any sequence, but 0(X) is a non-negative monotone function, we put (5.4) r fl = //X* for n = l,2,... h n The sets Y tl are measurable (96) on account of (4.1), and form an ascending sequence. We therefore have, by the part of our theorem proved alread), (5.5) (lim Y,,) = lim ( Y fl ). Now, it follows from (5.4) that Y n CX, and so, 0(Y n ) for each n. On the other hand, lim inf X n = lim ITn, and therefore n n the first of the relations (5.3) is an immediate consequence of (5.5). We establish similarly (or, if preferred, by changing X n to E X n ) the second of these relations, and this completes the proof of the theorem. Every function of a set <P( X), additive on a set E, can easily bo extended to the whole space X. In fact, if we write, for instance, S (JT)= $(X-E) for every set X measurable (9E), we see at once that 9i(X) is a function additive on the whole space JT, that co- incides, with $(X) for measurable subsets of E and vanishes for measurable sets containing no points of E. We shall call the func- tion &i(X), thus defined, the extension of <P(JT) from the set E to the space X. '10 CHAPTER I. The integral in an abstract space. $ 6. The variations of an additive function. The upper and lower bounds of the values that a function of a set $(Jt), ad- ditive on a set E assumes for the measurable subsets of this set E, will be called upper variation and lower variation of the function $ over E, and denoted by W((P;J5?) and W($;l?) respectively. Since every additive function vanishes for the empty set, we evidently have W(0;j0XO<W(0;JS). The number W(0;1B)+ |W((P; E)\ will be called absolute variation of the function <P on E and denoted by W(0;E). (6.1) Theorem. If $(X) is an additive function on a set E, it* variations over E are always finite. Proof. Suppose that W(#; E) = + oo. We shall show firstly that there then exists a sequence \E,^ n 1,2,. of sets (9E) such that (6.2) EnQE,,^ for ^>1; W(#;JS fl ) -oo; \9(E n )\^n l. For let us choose E l =^E and suppose the sets E n for n 1,2, ...,fc defined so as to satisfy the conditions (6.2). By the second of these conditions with n -- A;, there exists a measurable set A C E tt such that (6.3) \9(A)\^\V(E k )\ + lt. If W(</>; A) = oo, we have only to choose Ek+\~A in order to satisfy the conditions (6.2) for n=k + l. If, on the other hand, W(</>; A) is finite, we must have W(<P; #* A) = + oc, and, by (6.3), $(E k A)\^\<l>iA)\ ^(E^l^k, so that the conditions (6.2) will be satisfied for n = k+l, if we choose Ek+\ = E k A. The sequence (E n \ is thus obtained by induction. Now, on account of Theorem 5.1 and of the third of the con- ditions (6.2), we should have the equality and since every additive function of a set is, by definition, finite, this is evidently impossible. Q. E. D. It follows from the theorem just proved that every function $(X) additive on a set E is not only finite for the subsets (9E) of Ej but also bounded; in fact, the values it assumes are bounded in modulus by the finite number W(#; E). [ 6] The variations of an additive function. 11 Theorem 6.1 can be further completed as follows: (6.4) Theorem. For every function F(X) additive on a set E* the variations W($;JT), W(#;JT) and W(<P;.Y) are also additive functions of a set (96) on E, and we have, for every measurable set (6.5) Proof. To fix the ideas, consider the function Q^X) = W"(<P; X ). Since this function is finite by Theorem 6.1, we have to show that for every sequence {X n } of measurable sets contained in E, and such that X f X* = whenever i =^ ft, (6.6) QASXJ^SQ^X*). n n For this purpose, let us observe that for every measurable set XC,-X, t n we have 9(X) = S^(X-X,,)^SQ l (X >l ), and hence /i n (6.7) Q^SX H )^SQ^(X H ). n n On the other hand, denoting generally by 7,, any measurable set variable in X,,, we have Q l (2X,,)^^(2J tl ) = S9(J H ), and therefore also Q^SX^^SQ^X,,). Combining this with (6.7) we n n get the equality (6.6). Finally, to establish (6.5), we observe that for every measurable subset Y of X we have Q(J)=V(X) $(X Y)^0(Z) W(*;JT) f These two inequalities give together the equality (6,5), and the proof of Theorem 6.4 is complete. It follows from this theorem that every function of a set additive on a set i? is, on E, the difference of two non-negative additive functions. The formula (6.5) expresses, in fact, $(X) M the sum of two variations of $(X), of which the one is non-negative and the other non-positive; this particular decomposition of an additive function of a set will be termed the Jordan decomposition. 12 CHAPTER I. The integral in an abstract space. We can now complete Theorem 5.1 as follo'ws: (6.8) Theorem. If $(X) is additive on a set E, we have $(lim jr,,) = lim $(X n ) for every convergent sequence {JT,,} of sets (9) // n contained in E. In fact, making use of the Jordan decomposition, we may restrict ourselves to non-negative functions #(JT), and for these Theorem 6.8 follows at once from the second part of Theorem 5.1. S 7. Measurable functions. Given an arbitrary condition, or property, ( F) of a point x, let us denote generally by B[( V)] A' the set of all the points x of the space considered that fulfill this condition, or have this property. Thus, for instance, if f(x) denotes a function of a point defined on a set E and a is a real number, the symbol (7.1) E[xeE',f(x)>a] \~ denotes the set of the points x of E at which f(x) > a. A function of a point, /(#), defined on a set E, will be termed measurable (9E), or simply function (9E), if the set J5, and the set (7.1) for each finite a, are measurable (96). It is easy to see that (7.2) In order that a function f(x) be measurable on a measurable set E) it suffices that the set (7.1) should be so for all values of a be- longing to an arbitrary everywhere dense set R of real numbers (the same holds with the set (7.1) replaced by the set E[o?el?; / In fact, for every real a, the set R contains a decreasing sequence cf numbers (r/,5 converging to a. We therefore have oo E[a? e !?;/(#)> a] = J^E[#e E}f(x)>r ll ] and, each term of the sum .v n~ 1 x on the right being measurable by hypothesis, the same holds for the sum itself (cf. Theorem 4.1). Every function f(x) measurable on a set 15, can be continued in various ways, so as to become a measurable function on the whole space X. For definiteness, we shall understand by the ex- tension of the function f(x) from the set E to the space Jf, the func- tion / (a?) equal to f(x) on E and to everywhere else. For brevity, we shall often deal only with functions measurable on the whole If 7] Measurable functions. 13 space X 9 but it is easy to see that all the theorems and the reasonings of this and of the succeeding, could be taken relative to an arbitrary set (9E). The equations a] = CE[/(a?) > a], E [/(a?) ^ a] = /7 E \f(x) > a - .r A /i-l A L a] = CB [/() ^ a], E[/(a?)=a] = = E[/(ar)<n], E[/(a>) > oo] = E[f(x) > n], II 1 V E[/(a>)=+oo]=CE[/(a!)<+oo], E[/(ar) = oo] = CE [/()> ex.] A' .V V A show that for every measurable function f(x) and for every number a, the left hand sides are measurable sets. Conversely, in the definition of measurable function, we may replace the set (7.1) by any one of the sets E [/(#)> a], E[/(#)^a}or E [/(#)< a]; this follows at A' A A' once from the identity a] = I Ef/(a?)^a+^| - CE[/(a?) ^a] = C/l E To any function f(x) on a set E, we attach two functions f(x) and /(a?) on 1?, called, respectively, the non-negative part and the non-positive part of /(#) and defined as follows: or according as f(x) ^ or f(x) < 0, f(x) = f(x) or according as f(x) < or /(#) ^ 0. o We see at once, that in order that a function be measurable on a set J?, it is necessary and sufficient that its two parts, the non-negative and the non-positive, be measurable. Eeturning now to the notions of characteristic function, and simple function introduced in 3, we have the theorem: (7.3) Theorem. In order that a set E be measurable (90, it is ne- cessary and sufficient that its characteristic function be measurable. More generally , in order that, on a set E, a simple function f(x) be measurable (9), it is necessary and sufficient that, for each raluc of /(a?), the points at which this value is assumed on E, should con- stitute a measurable subset of E. Another theorem, of groat utility in applications, is the fol- lowing: 14 CHAPTER 1. The integral in an abstract space. (7.4) Theorem. Every function f(x) that is measurable (9E) and non-negative on a set E, is the limit of a non-decreasing sequence of simple functions, finite, measurable and non-negative on E. In fact, if we write for each positive integer n and for n, if f(x)^n, the functions /(#) thus defined are evidently simple and non-negative, and, on account of Theorem 7.3, measurable on E. Further, as is easily seen, the sequence !/(#)) is non-decreasing. Finally lim/ /l (a?)==/(o?) n for every xeE; for, if f(x) <+oo, we have, as soon as n exceeds the value of f(x), the inequalities O^f(x) f n (x) ^ 1/2", while, if f(x) = -f oo, we have //,(#) = n for n = 1, 2, ..., and so $ 8. Elementary operations on measurable functions. We shall now show that elementary operations effected on measur- able functions always lead to measurable functions. (8.1) TheorviM. Given two measurable functions f(x) and g(x), the sets E [/(*)> jr(*)l, E [/(*) are measurable. The proof follows at once from the identities E \l(x) > g(x)] = + f + f E [/(*) > E g(x) < , A w oo m \ .v I m E[/>jf] = CE[</>/] and E[/ = g] = A A' A 1 A" A* (8.2) Theorem. If the function f(x) is measurable, \f(x)\" is also a measurable function. For a > 0, the proof is a consequence of the identity E[|/(a?)|" > a] = E[/(0) > a 1 ] + E[/(a< a 1 ], X X Jt which is valid for every a ^ 0, while for a < its left hand side coincides with the whole space and therefore constitutes a mea- surable set. For a < 0, the proof is similar. [jj 8] Elementary operations on measurable functions. 15 (8.3) Theorem. Every linear combination of measurable functions with constant coefficients represents a measurable function. The identities A^ /* E [a. /(afl + 0> a] = E .a for a > 0, for a < 0, valid for every function f(x) and for all numbers a, a ^ 0, and fi, show, in the first place, that a. f(x) + p is a measurable function, if f(x) is measurable. It follows further, from Theorem 8.1 and from the identities: that if f(x) and g(x) are measurable functions, so is a /(#) + /i g(x). (8.4) Theorem. The product of two measurable functions f(x) and g(x) is a measurable function. Measurability of the product f-g is derived by applying Theorems 8.2 and 8.3 to the identity fg = [(/ + g) 2 (/ gf) a ], the com- pletion of the proof, by taking into account possible infinities of / and g, being trivial. (8.5) Theorem. Given a sequence of measurable functions (/(#)!, the functions upper bound/,, (x), lower bound ///(a?), limsup/,,(#) and liminf/,,(#) /; n n n are also measurable. The measurability of h(x) = upper bound f,,(x) follows from // the identity E[h(x) >a]= 2Tb[f n (x)>a]. For the lower bound, the corresponding proof is derived by change of sign. Hence, the functions A,,(#)=upper bound [//, > j (a?), /, + >(#)> ] are measurable, and the same is therefore true of the function limsup fn(x) = lim h n (x) = lower bound h,,(x). By changing the sign n of /i(x), we prove the same for lim inf. 16 CHAPTER I. The integral in an abstract space. 9. Measure. A function of a set n(X) will be called a meas- ure (3E), if.it is defined and non-negative for every set (9E), and if for every sequence {X,,} of sets (9E) no two of which have points in common. The number p(X) is then termed, for every set ^mea- surable (SE), the measure (p) of X. If every point of a set Ej except at most the points belonging to a subset of E of measure (/*) zero, possesses ft certain property F, we shall say that the condition V is satisfied almost everywhere (p) in J5, or, that almost every (fi) point of E has the property V. We shall suppose, in the sequel of this Chapter, that, just as the class 9E was chosen once for all, a measure n corresponding to this class is also kept fixed. Accordingly, we shall often omit the symbol (fi) in the expressions "measure (/*)", "almost everywhere (fi)", etc. Clearly //(T)^//(Y) for any pair of sets X and F measurable (9E) such that X(2 ^> an( i for every sequence of measurable sets \X n }. A measure may also assume infinite values, and is therefore not in gen- eral an additive function according to the definition of 5. The results established in this chapter concerning perfectly arbitrary meas- ures \vill be interpreted in the sequel for more special theories of measure, (for instance, those of Lebesgue and Taratheodory). For the present, we shall be content mentioning a few examples. Let us take for 9C, the class of all sets in a space A'. We obtain a trivial example of measure (9E) by writing //(AT)^O identically, (or else //(JC)^-f-oo) for every set X(^A'. Another example consists in choosing an element a in A" and writing /*(JT) 1 or /(JT)=^0, according as aeX or not. In the case of an enumerable space A', consisting of elements a l9 a 2 , ...,#/* ..., the general form of a measure ft(X) defined for all subsets X of A" is ji(X)= 2k n fn(X) where /? {kn} is a sequence of non-negative real numbers and fn(X) is equal to 1 or ac- cording as a t ,X or not. It follows that every measure defined for all subsets of an enumerable space, and vanishing for the sets that consist of a single point, vanishes identically. The similar problem for spaces of higher potencies is much more difficult (see S. HI a in fl]). For a space of the potency of the continuum see also S. Banach and C. Kuratowski [1], E. Szpilrajn fl], W. Sierpinski [I, p. 60], W. Sierpinski and E. Szpilrajn [1J. We shall now prove the following theorem analogous to The- orem 5.1: (9.1) Theorem. If {X,,} is a monotone ascending sequence of measurable sets, then lim/i(X n ) = /*(lim X,,). The mme holds n n monotone descending sequences provided, howe^^er, that [f 9] Measure. 17 More generally, for every sequence {X n } of measurable sets, (9.2) it (lim inf X H ) < lim inf it (X n ) n n and, if further n(X n )-[ oo, n (9.3) /< (limsupJT,,) 5* lim sup //(A',,), n n so that, in particular, if the sequence {X n } converges and its sum has finite measure, lim n (X n ) = j* (lim X n ). n n Proof. For an ascending sequence {Zn} B =i, f ... the equation lim/'(X,)==/<(limXi) follows at once from the relation n n OO 00 === JL.II == J*-\ \ 2- \J*-n \-\ "~~ JLn), n^\ n 1 and if the sequence {X n } is descending and /ifJS^J^oo, then the measure n(X) is an additive function on the set X^ and consequently the required result follows from Theorem 5.1. In exactly the same way, if for an arbitrary sequence [X n ] of measurable sets, ^ X fl is of finite measure, the measure p(X) n is an additive function on this set, and the two inequalities (9.2) and (9.3) follow from Theorem 5.1. To establish the first of these inequalities without assuming that the sum of the sets X has finite measure, we write as in the proof of Theorem 5.1 Since the sequence is ascending, arid Y,,CTA%, for every n, we have /* (lim inf X,,) == ^(lim Y,,) = lim ju( ) <:. lim inf /i (X n ). n n n n We conclude this with an important theorem due to D. Ego- roff, concerning sequences of measurable functions (cf. D. Ego- roff [11, and also W. Sierpinski [3], F. Eiesz [2; 3], H. Hahn [I, pp. 556 8]). We shall first prove the following lemma: (9.4) Le-Hima. If E is a measurable set of finite measure (u) and if {///(#)) w a sequence of finite measurable functions on E, con- verging on this set to a finite measurable function /(#), there exists, for each pair of positive numbers f , fj, a positive integer N and a mea- surable subset H of E such that n(H)<t] and "(9.5) \fn(x) f(x)\<f for every n>JV and every xeE H. 18 CHAPTER I. The integral in an abstract space. Proof. Let us denote generally by E m the subset of E con- sisting of the points x for which (9.5) holds whenever n > m. Thus defined, the sets E m are measurable and form a monotone ascending sequence, since for each integer w, we have E m = ft E[x E;\f(x)-fn(x)\<e]. n-~m-f t jc Further, since {/*(#)) converges to f(x) on the whole of E, we have E=s%E m , and so, by Theorem 9.1, ^($) = lim^($ m ), i. e. m m limp(E E m ) = Q, and therefore, from a sufficiently large m in onwards, n(E E m )<*i. We have now only to choose N~m and H=E J,, lo , and the lemma is proved*. (9.6) EyoroJFs Theorem. If E is a measurable set of finite measure (fi) and if (fn(x)} & a sequence of measurable functions finite almost everywhere on E, that converges almost everywhere on this set to a finite measurable function /(#), then there exists, for each e>0, a subset Q of E such that ft(E (?)< and such that the converg- ence of {fn(x)} to f(x) is uniform on Q. Proof. By removing from E, if necessary, a set of measure (ft) zero, we may suppose that on J5, the functions f n (x) are everywhere finite, and converge everywhere to f(x). By the preceding lemma, we can associate with each integer m>0 a set H m (2E such that p(H m ) < f/2' 11 and an index N m such that (9.7) \f n (x) f(x) | < 1/2'" for n > N m and for xe E H tn . oo Let us write Q = J5 v JJ WM We find //r-i // (EQ) ^ 1 /*(ffm) < 5 /2"' = f, in 1 and since the sequence /(#) converges uniformly to f(x) on the set Q on account of (9.7). the theorem is proved. [| 10] Integral. 19 The theorem of Egorof f can be given another form (cf. N. Lusin [I, p. 20]), and, at the same time, the hypothesis concerning finite measure of E can be slightly relaxed. (9.8) If E is ike turn of a sequence of measurable sets of finite measure (/<) and if {fn(x)} is a sequence of measurable functions finite almost everywhere on this set, converging almost everywhere on E to a finite function, then the set E can be expressed as the sum of a sequence of measurable sets H, E lt 2 s , ... such that f*(H) and that the sequence (1n(x)} converges uniformly on each of the sets En. For the proof, it suffices to take the case in which the set E is itself of fin- ite measure. With this hypothesis, we can, on account of Theorem 9.6, define n by induction a sequence [Ek}k=i,2,... of measurable sets such that /* (E ^Ek) ^ 1/n, and that the sequence {f n (x) } converges uniformly on the set Eh for each fc. Choosing 00 H E y En, we have u(H)= 0, and the theorem is proved. *=i As we may observe, the hypothesis that the set E is the sum of a sequence of sets of finite measure, is essential for the validity of Theorem 9.8. For this pur- pose, let us take as a space Jf the interval [0, 1], and as an additive class SE of sets, that of all subsets of X^. Further, let us define a measure /i by writing /w (J) = oo whenever the set X c 9f is infinite and ^ (^) = n if JT ia a finite set and n denotes the number of its elements. The sets of measure (<w ) zero then coincide with the empty set. Finally, let (gn(x)} be an arbitrary sequence of fun- ctions, continuous on the interval [0, 1], converging everywhere on this interval, but not uniformly on any subinterval of [0, 1]. To justify our remark concerning Theorem 9.8, it suffices to show that the interval X = [0, 1] is not representable as the sum of a sequence {En} of sets such that the sequence of functions (gn(x)} converges uniformly on each of them. But if such a decomposition were to exist, we might suppose firstly since the functions gn (x) are continuous all the sets En closed. Then, however, by the theorem of Baire (cf. Chap. II, 9) one of them at least would contain a subinterval of [0, 1]. This gives a contradiction, since by hypothesis, the se- quence (gn(x)} does not converge uniformly on any interval whatsoever. 10. Integral. If we are given in the space X an additive class of sets 9E and a measure p defined for the sets of this class, we can attach to them a process of integration for functions of a point. In fact: (i) If f(x) is a function (3E) non-negative on a set E, we shall understand by the definite integral (9E, /*) of f(x) over E the up- per bound of the sums n where {Ek}k=\,t ..... n is. an arbitrary finite sequence of sets (9E) such that E=E l + Ei+... + E n and E t E k = for i + fc, and where V*, for &=1,2, ...,n, denotes the lower bound of f(x) on E k . 20 CHAPTER I. The integral in an abstract space. (ii) If j(x) is an arbitrary function measurable (9E) on a set J5, we shall say that f(x) possesses a definite integral (9E, /*) over E, if o one at least of the non-negative functions f(x) and f(x) (cf. 3) o possesses a finite integral over E according to definition (i). And, if this condition is satisfied, we shall understand by the definite integral (9E> //) of the function f(x) over E the difference between the integral of f(x) and that of f(x) over E. The definite integral (9E, p) of f(x) over E will be written (9E) / f(x) dp (a?). If this integral k is finite, the function f(x) is said to be integrable (96, /*). For every function f(x) possessing a definite integral over a set J5, we evi- dently have = (*) //>-(*) /'(/)** = () We see at once that the two definitions (i) and (ii) are com- patible, i. e. that they give the same value of the integral to any non-negative measurable function. Moreover: (10.1) If g = (v l9 XrfVij X 2 ; ...;t>, 7J , X m \ is a simple non-negative func- tion on the set E = X l + X% + + X m > the sets Xj being measur- able (9E), then For, if {Ej}j i,>, ,,, is an arbitrary subdivision of E into a finite number of sets (9E) without points in common, and if w/ denotes the lower bound of g(x) on E n we have w t . v t whenever E { - Z/ 4= 0. Hence V y-i y i / i y--i / i / i and therefore f gdu^^ViftiX,). The opposite inequality is ob- K ~ <+, vious, since the sets J n Z 2 ,..., J m themselves constitute a subdivision of E into a finite sequence of sets (9E) on which the values of g(x) are v^v 2 j...jO in respectively. [ 11] Fundamental properties of the integral. 21 11. Fundamental properties of the integral. We shall begin with a few lemmas concerning integration of simple functions. As in the preceding , the symbols 9E, ^ etc. will often be omitted. (11.1) Lemma. 1 For every pair of functions g(x) and h(x), simple, non-negative, and measurable (9E) on a set E, we have (11.2) f[g(x) + h(x)]dt*(x) = fg(x)d f i(x) + fh(x)d/4(x). K E E 2 // the function f(x) is simple, non-negative, and measurable (9E) on the set A + B where A and B are sets (9E) without common points, then (11.3) ff(x)dn(x) = ff(x)dit(x) + ff(x)d f i(x). A+B A B Proof. As regards 1, let 9 = (9u 0i J 02> *> 5 9> G "} and h = (fc t , jB\; h 2 , H 2 ] ...; h m , H m } f where E = G L +...+G H = H^^.+H.n. We then have, by (10.1), / [g(x)+h(x)] dii (x) = V v ( g. +hj ) p (G . . Hj ) = As regards 2, if i7 = ^+J8 and / = |/ lf where ^=^+^2+ +Q/n we have + (11.4) Lemma. If {^,,(0?)} i a non-decreasing sequence of functions that are simple, non-negative, and measurable (9E) on a set J3, and if, for a function h(x), simple, non-negative, and measurable, on E, we have lim g n (x) ^ h(x) on E, then n (11 .5) lim / g n (x) d.u (x) ^ / A(x) ri/< (x). fi ^ . 22 CHAPTER I. The integral in an abstract space. Proof. Let * = {% E^ t? 2 > ^a> } v m> E m}j where 0< i < t> t < ... <v m and E =^E l +E 2 + . We may suppose v l > 0, for, otherwise, we should have J hdp = j hdfij and, since f g n dp:^ f g n dp, we could replace E E~E I E E-EI the set E by the set E E l on which h (x) does not vanish any- where. Further we shall assume first that v m <. + . Let us choose an arbitrary positive number e < v^ and let us denote, for each positive integer n, by Q n the set of the points x of E for which g n (v)>h(x) * The sets Q n evidently form an ascending sequence converging to J5, and, by Theorem 9.1, we have P(Qn)-+l*(E). This being so we have two cases to distinguish: (i) i*(E) 4= - We then can find an integer n such that for n>n we have n(E Q n ) < *, and therefore, by Lemma 11.1, f [h(x) Q n E K and, passing to the limit, making first n -> oc, and then e -> 0, we obtain the inequality (11.5). (ii) p(E)=oo. Then, sinc^ I g n dii'^(v l e)n(Q tl ), we obtain E Uni fg n dn=oo, so that the inequality (11.5) is evidently satisfied. n J E Suppose now v m = + . Then by (10.1) and by what has m-J already been proved, lim / g n dfi^v^(E m )+ Vrn(Et) for any n & i^l finite number t;, and consequently for v = + oo = v m also; whence, in virtue of (10.1) the inequality (11.5) follows at once. (11.6) Lemma. If the functions of a non-decreasing sequence (gn()} we simple, non-negative, and measurable (SE) on a set E, and if g(x) = lim g n (x), then lim / g n (x) d t u (x) = / g(x) dp (x). " E E Proof. Let E l ,E 2 ,...,E m be an arbitrary subdivision of E into a finite number of measurable sets, and let v l9 v 2 , ..., v m be the lower bounds of g(x) on these sets respectively. Let us write t?={t? 1 , Erf v 2 , J57 2 ; ...; v m , E m }. We evidently have lim g n (x) = n on E, and hence, by Lemma 11.4 and by Theorem 10.1 [{ 11] Fundamental properties of the integral. 23 lim I' g n dfi ^ f v dfl = git, fi (Et). " E K /=1 It follows that Hmfgndu^ f gd t u, and since the opposite E E inequality is obvious, the proof is complete. We are now in a position to generalize Lemma 11.1 as follows: (11.7) Theorem. The relation (11.2) holds for every pair of functions, g(x) and h(x), non-negative and measurable (9E) on the set E, and the relation (11.3) holds for every function f(x) non-negative and measur- able (SE) on the set A + B, where A and B are sets (9E) without points in common. Proof. By Theorem 7.4 there exist two non-decreasing se- quences (</(#)) an( J iM#)} of simple non-negative functions mea- surable (9E) on E, such that g(x) = ]img n (x) and h(x) = \imh n (x). n n Now, by Lemma 11.1 (1), we have f (g n + h fl )dft ==. f g n dft -f j h n du E E E and hence, making n -> oo, we obtain, on account of Lemma 11.6, the relation (11.2). Similarly, if we approximate to f(x) on A + B by a non- decreasing sequence of simple non-negative functions and make use of Lemma 11.1 (2), we obtain the relation (11.3). (11.8) Theorem. 1 For any junction measurable (9E), the integral over a set of measure zero is equal to zero. 2 // the functions g(x) and h(x) measurable on a set E are almost everywhere equal on E, and if one of the two is integrable on E, so is the other, and their integrals over E have the same value. 3 // a function f(x) measurable (9E) on a set E has an integral over E different from +00, the set of the points x of E at which f(x) -f has measure zero. In particular, if the integral of f(x) over E is finite, the function f(x) is finite almost everywhere on E. Proof. We obtain at once part 1 of this theorem by making successive use of the definitions (i) and (U) of 10. As regards 2, it is evidently sufficient to consider the case of non-negative functions g(x) and h(x). If we denote by E l the set of the points x of E at which g(x)= h(x), we ha\e by hypothesis 1 ) = 0, and, on account of (1) and of Theorem 11.7, we obtain J gdp == I gdfi = I hdp = / ftd.w, as required. K -#, K 24 CHAPTER I. The integral in an abstract space. Finally, as regards 3, let us suppose that for a function f(x) measurable (9E) on E we have f(x) = + on a set E Q (^E ot positive measure. We then have / f dp ^ ] f dp^n-n (E Q ) for every n, and E K {> so | / d/ = -|~ oo. Consequently, the integral of f(x) over E, if it /: exists, is positively infinite, and this completes the proof. We now generalize Lemma 11.1 (1) and also complete The- orem 11.7, as follows: (11.9) Theorem of distributivity of the integral. Every linear combination with constant -coefficients, a*g(x) + b'h(x) of two func- tions g(x) and h(x), integrablc (3E, ^) over a set E, is also integrable over E, and we have (11.10) l(ag + bh)du=a f gdp + b j hdp. K K K Proof. By Theorem 11.8 (3), the set of the points at which either of the functions g(x) and h(x) is infinite, has measure zero, and if we replace on this set the values of both functions by 0, we shall not affect the values of the integrals appearing in the relation (11.10). We may therefore suppose that the given functions g and h are finite ori E. Further, the relations / agdp = a gd<t, I bhd}* = b hdu E E K K being obvious, we need only prove the formula (11.10) for the case a=ft=l. Finally, the set E can be decomposed into four sets on each of which the two functions g(x) and h (x) are of cons- tant sign. So that, on account of Theorem 11.7, we may assume that the functions g(x) and h (x) are of constant sign on the whole set E. Now, by the same theorem, the relation (11.11) f(g + h)du=fgdn+ fhdu holds whenever the functions g and h are both non-negative or both non-positive on E 9 and it only remains, therefore, to show that this relation is valid when g and h have, on E, opposite signs, the one, g(x) say, being non-negative, the other, h(x), non-positive. [ 11} Fundamental properties of the integral. 25 This being so, let E l and U 8 be the sets consisting of the points x of E for which we have g(x) + h(x)^ and g(x)+ h(x)<Q, respectively. The functions 0, g+h, and h are non-negative on E l and we therefore have, by Theorem 11.7, fgdi*= f(g + K)dp+ f(h)d<< = f(g+h)dn fhd?<. A', A, K, A;, k t Similarly k, A ; , /;, X*. k. E. Therefore, for i = l, 2, we have I (g+h) di* = J gdp + J hdp, and k t A, A, by Theorem 11.7 we obtain the relation (11.11). (11.12) Theorem on absolute inte(/ral>ility. 1 In order that a function f(x) measurable (9E) on a set E should be integrable (9E, /*) on E, it is necessary and sufficient that its absolute value should be so. 2 I/, for a function g(x) measurable (SE) on a set E, ti. e exists a function h(x), integrable (X, /*) and such that \g.(x)\'^h(x) on E, then the function g(x) also is integrable on E] in particular, every function measurable (9E) and bounded on a set E of finite measure (fi) is integrable (96, /O on E. Proof. As regards 1, we have by Theorem 11.7 /'i/i dp = /';** ' and integrability of |/| is therefore equivalent to that of / and that of / holding together, i. e. to integrability of /. As regards 2, we have the inequalities </(a?)^|sr(a?)|^ft(#) and 9(x) ^ \g()\ ^ M#) n ^> aad, since h(x) is, by hypothesis, inte- grable on Ej it follows that the same is true of the non-negative func- tions g and 0, and therefore of the function g(x). 26 CHAPTER I. The integral in an abstract space. As an immediate consequence of Theorem 11.12 we have the following theorem, known as the (11.13) First Mean Value Theorem. CKven, on a set JE, a function f(x) bounded and measurable (9E) on E and a function g(x) integrable (9E, (*>) on E, the function f(x) g(x) is integrable on E and there exists a number y lying between the bounds of f(x) on E, such that (11.14) f\1(x)\g(x)\dp(x) =y- f\g(x)\dfi(x). E E Proof. If we denote by m and M respectively the lower and the upper bound of f(x) on E, and make use of Theorem 11.12, we verify successively, that the functions (\M\+\m\)-\g(x)\, \f(x)g(x)\, f(x)\g(x)\ and f(x) g(x) are integrable on E. Further, we have m \9( x )\ ^ 1( x )'\g( x }\ ^ M\g(x)\ over E, and, therefore also, m I \g\du^ I f-\g\dn^M /|jf|<fy*, and so choosing y=\ f f-\g\dfi\:\ f \g\dfi] E E E E E (or, if the denominator vanishes, an arbitrary y between m and M ), we obtain the formula (11.14) with m 12. Integration of sequences of functions. In this , we shall establish some theorems on term by term integration of sequences and series of functions. (12.1) Theorem. If the functions of a sequence (g n (x)} are finite and integrable (9E, //) on a set E of finite measure, and the sequence converges uniformly on E to a function g(x), then the function g(x) also is integrable over E, and we have (12.2) Um [g H (x)diA(x)= (g(x)dn(x). " k E Proof. By Theorem 8.5, the function g(x) is measurable (9E) on E. The functions g(x) g tl (x) are therefore all measurable also, and, further, since the sequence [g n (x)} converges uniformly to g(x) the functions g(x) g n (x) are all bounded, at any rate from some value of the index n onwards. These functions are thus, by The- orem 11.12 (2), integrable on E 9 and it follows, by Theorem 11.9, that the function g (x) = [g (x) g,,(%)] -f <///(#) is integrable too. Finally, denoting by e,, the upper bound of \g(x) <M#)| on ^> we have 15 J-2Ji Integration of sequences of functions. 27 and this establishes the relation (12.2) since, by hypothesis, *->() and ft (E) 4= - Neither the theorem thus established, nor its proof, contains, at bottom, anything new, as compared with the similar result for the, classical processes of integration of Cauchy, or of Riemann. V*\> now pass on to th^, proof of theorems more closely related to Lebesgue integration. Among these theorems, a fundamental part is played by the following one,, which is due to Lebesgue: CO (12.3) Theorem. Let f(x) ^ f n (x) be a series of non-negative functions n i measurable (i?) on a set E. Then * (12.1) I f(x)dp(x)=v I / H ( X )dp(x). K "- } K Proof. From Theorem 11 7, we derive in the first place, that /' /' "' '" / / jdii r^ I {V f tl <iu\ ^= v I f n ,i fl f or every w, and so k K " ] " ! A- CXJ '-'-*) I fdp>2 I fndfi. h " } K To ostal>lish the opposite inequality, let us attach, in accordance with Theorem 7.4, to each function f n (x) a non-decreasing sequence {g ( n\$)}ti 1.2. ( >f simple functions measurable and non-negative on B, in such a manner that Urn g ( } (x) = f () for n=l, 2,... Let us write k (a;)=v (j\ ft) (x). The functions x k (x) are clearly simple, measurable, / i and non-negative, on E, and they form a non-decreasing sequence. m Further, for each w, and for k ^m, we have ^ g\ (x) ^ s k (x) <: /(a?). / in Making k >oo, we derive Jf f,(x) ^ lim s k (#) ^f(x) for every m, i 1 A and so, /(.*) = lim **(#). Therefore, by Lemmas 11.6 and 11.1 (1), h I fdn = lim ft* dp - lim v j'^dp ^ v // |d/s K * K k ' * K '' ' fc and this, combined with (12.5), gives the equality (12.4). 28 CHAPTER I. The integral in an abstract space. Theorem 12.3 may also be stated in the following form: (12.6) Lebcsffue's Theorem- on integration of monotone se- quences of functions. If \f n (x)} is a non-decreasing sequence of non-negative functions measurable (9E) on a set E, and /(#) = lim /(#)> then I f(x) dn (x) = lim / /(#) dti (a?). /; " k Proof. If we write g,,(x) = /,,-n //i(#), we obtain and the functions g n ($) will be non-negative and measurable on E, so that by Theorem 12.3 / fdu = / M/" -f 5' / 9nd = lim / [f l f ^g n ]dft = lim/ f k d. ' /: Q. E. D. (12.7) Throrwn of ailditlvity for the Integral. If {E n } is a sequence of sets measurable (9t) no two of which have common points, and E=^iE n j then n (12.8) ffdf* = V I'fdn H K n for every junction f(x) possessing a definite integral (finite or infinite) over E. Proof. It is clearly sufficient to prove (12.8) in the case of a, function f(x) non-negative on E. Supposing this to be the case, let us write f n (x) = f(x) for xeE,,, and /() = for xt-E E n . We then have f(x)^^f n (x) on E, and, the functions / being measurable /; and non-negative, we may apply Lobogguo's Theorem 12.3. This gives, by Theorem 11.7, [fdu - V / fn dp ^ V [ fn dn -v [ fd p m Q. B. D. i, " K K H " /: It' a function. f(x) has tt definite integral (,/<) over a set E, then f(x) also has a definite integral over any subset of E mea- surable (9E). We may therefore associate with it the function of a set (9E) defined as follows: (lli.9) F(\) = f f(x)<Ifi(x) where XC E and JTt*. [ 12] Integration of sequences of functions. 29 The latter will be called the indefinite integral (SE, /*) of f(x) on E. It follows from Theorem 12.7 that, whenever the function f(x) is integrable (9E, //) on E, its indefinite integral is an additive func- tion of set (9E) on E. We end this with two simple but important theorems. The first is known as Fatou's lemma, and appears for the first time (in a slightly less general form) in the classical memoir of P. Fa to u [1, p. 375] on trigonometric series. The second is due to Lebesgue [5, in particular p. 375], and is called the theorem on term by term integration of sequences of functions', cf. also Ch. J. de la Valise Poussin [1, p. 445453], R. L. Jeffery [1] and T. H. Hilde- brandt [2]. (12.10) Theorem ( Fatou's Levnma). If \f n (x)} is any sequence of non-negative functions measurable (9E) on a set E, we have I lim inf /(#) d'i (x) <; lim inf / f n (x) dn (x). ' Proof. Let us write Sf/(a7) = lowerbound[//(a7),/ /+ i(a?),/^ 2 (a?), ...] where t=l, 2, .... Thus defined [g,(x)\ is a non- decreasing sequence of non-negative functions measurable on E, and converges on the set .E to lim inf //(#) We therefore have, by Lebesgue's Theorem 12.6, i /lim inf f,(x)du(x) = lim f g, (x) du (x) ^ lim inf f f,(x)dp(x). K ' E ' E (12.11) Lebesguc'H Theorem on term by term integration. Let {f,,(x)} be a sequence of functions measurable (9E) on a set E, fulfilling, for a function s(x) integrable (9E) on JS, the inequality \1,,(x)\ <: s(x) for n=l, 2, ... Then fn d > lim inf f n dp. (12.12) * lim sup f f n dfi ^ f lim sup / du. " .E K I/, further, the sequence {f,,} converges on E to a function f, the sequence is integrable term by term, i. e. we have (12.13) lim ff n dn 30 CHAPTER I. The integral in an abstract space. Proof. Let g(x) = lim inf f n (x) and let h(x) = lim sup /(#). n n We may clearly suppose s(x) < + oo throughout E. We then derive from Fatou's Lemma 12.10, lim inf (* + /n)<fy*> (* + g)dp and E B lim inf f (s f n )dp^* [ (s h)dp, which gives at once the rela- " B E tions (12.12). Further, if lim f n (x)= /(a?), we derive from (12.12) the relation n lim inf f f n dp > / fdfi ^ lim sup f f n dp which gives the equality E E n E (12.13). 13. Absolutely continuous additive functions of a set. The fact that the indefinite integral of a function integrable ($, p) on a set E is, on E, an additive function of a set (SE), raises the problem of characterizing directly the additive functions expres- sible as indefinite integrals. If we restrict ourselves to the Lebesgue integral of functions of a real var- iable, we may regard indefinite integrals as functions of an interval, or, what comes to the same thing, as functions of a real variable. In that case, a neces- sary and sufficient condition for a function to be expressible as the indefinite integral of a real function was given, in 1904, still by Lebesgue [I, p. 129, foot- note]. A little later (in 1905), G. Vitali [1] explicitly distinguished the class of functions possessing the Lebesgue property by introducing the name of "ab- solutely continuous functions". The condition of Lebesgue and Vitali was later extended to functions of A set by J. Radon [1] (cf. also P. J. Daniell [2]). But Radon considered only additive functions of sets measurable in the Borel sense in Euclidean spaces, and only measures determined by additive functions of intervals (cf. below Chap- ter HI). The final form of the condition of Lebesgue- Vitali, as given in Theorem 14.11 below, is due to 0. Nikodyra [2]. An additive function of a set (9E) on a set E, will be said to be absolutely continuous (9E, p) on E, if the function vanishes for every subset (9E) of E whose measure (p) is zero. An additive function $(X) of a set (9) on a set E will be termed singular (9E, p) on JEf, if there exists a subset E Q C.E measurable (9E), of measure (p) zero, such that $(X) vanishes identically on E JE/ , i. e. 0(X)=0( -X) for every, subset X of E measurable (9E). The following statements are at once obvious: [ 13] Absolutely continuous additive functions of a set. 31 (13.1) Theorem. 1 In order that an additive function of a set (9E) on a set E should be absolutely continuous (9, p) [or should be singular] it is necessary and sufficient that its two variations, the upper and the lower, should both be so. 2 Every linear combination, with con- stant coefficients, of two additive functions absolutely continuous [or singular] on a set E is itself absolutely continuous [or singular] on E. 3 ty a sequence {<P n (X)} of additive functions, absolutely continuous [singular] on a set E, converges to an additive function $(X) for each measurable subset X of E, then the function $(X) is also absolutely continuous [singular]. 4 If a function of a set (9) is additive and absolutely continuous [singular] on a set E, the function is so on every measurable subset of E. 5 // E=^E n , where {E n } is a sequence of sets (9E), and if an additive function on E is absolutely continuous [singular] on each of the sets E n , the function is absolutely continuous [singular] on the whole set E. 6 An additive function of a set cannot be both absolutely conti- nuous and singular on a set 7?, without vanishing identically on E. For sets of finite measure, it is sometimes convenient to apply the following test for absolute continuity: (13.2) Theorem. In order that a function (X) additive on a set E of finite measure, be absolutely continuous (9E, /') on E, it is necessary and sufficient that to each >0 there correspond an i?>0, such that imply |<P(JT)|< for every set X(^E measurable (9E). Proof. It is evident that the condition is sufficient. To prove it also necessary, let us suppose the function $(X) absolutely con- tinuous in E. We may assume, replacing if necessary, $(X) by its absolute variation, that $(X) is a non-negative monotone func- tion on E. This being so, let us suppose, if possible, that there exists a sequence {/} 1,2,.. of measurable subsets of E, such that (j(E n )<II2" and that ^(E n )>%, where i? is a fixed positive number. Let us write E Q = lim sup E n - For every n, we then have ^l/ 2 ' 1 " 1 * and therefore f*(# )-0. On the other k n hand, by Theorem 5.1, we have <P(J5 )>lim sup $(E n )^r] . This /i is a contradiction, since $(X) is absolutely continuous, and the proof is complete. 32 CHAPTER I. The integral io an abstract space. (13.3) Theorem. In order that a function <D(X), additive on a set E, be singular (9E,ji) on J0, it is necessary and sufficient that for each >0 there exist a set XE measurable (9E) and fulfilling the two conditions fi (Z) < *, W( 0; E X)< e. Proof. The condition is clearly necessary. To prove it suf- ficient, let us suppose that for each n there is a set X n _E meas- urable (9E) such that /f(Z M )<l/2 fl and W(#; E X,)<l/2", and 00 let us write E g = lim sup X,,. We then have ft(E Q ) for each n, and so ^(J5 ) = 0. On the other hand, by Theorem 5.1 we have W(#; E j )<:iiminf W(<P; jB X,,) = 0. The function /i <P(X) is therefore singular on E. # 14. The Lebesgue decomposition of an additive function. Before proving the result announced in the preceding , we shall establish some auxiliary theorems. We begin with the follow- ing theorem due to H. Hahn [I, p. 404] (cf. also W. Sierpiiiski [11]): (14.1) Theorem. If <P(Z) is an additive function of a set (96) on a set E, there exists always a set P(^E measurable (9E), such that W(<P; P) = = W (<P; E P), or, what comes to the same thing, such that $(X)^*Q for every measurable set X(^P and <P(Z)<0 for every measurable set X C E P. Proof. For each positive integer n, we choose a set E n such that $(En)^W($;E) l/2". By Theorem 6.4 we then have, (14.2) W(0; E n ) ^ 1/2" and W(<P; E E n ) ^ 1/2". oo Writing P=liminf $, we see that E P=lim aup(E E n )C2(E &*) n n - for every w, and therefore, by (14.2), >m -I which gives W($;E P) = 0. On the other hand, the lower var- iation W((P; X) is a non-positive monotone function of a measurable set X(^E, and, by Theorem 5.1 and the first inequality (14.2), we must have the relation |W( <P; P)| < lim inf | W( (P; E n ) = 0, which n gives W(<P;P) = and completes the proof. [ 14] The Lebesgue decomposition of an additive function. 33 (14.3) Lemma. If (X) is a non-negative additive junction of a set (9) on a set E, there exists, for each a > 0, a decomposition of E into a sequence of measurable sets without common points, HjEuEu ...,J5 n , ... such that fi(H) = and that (14.4) a (n 1) fi( X) for every set X C E n measurable (9E). Proof. By Theorem 14.1, there exists, for each positive in- teger n, a measurable set A n such that $(X) an-fi(X)^0 'for every measurable set X(2A n and (X) an ^ (X) ^ for every CO measurable set XE A n . Write B n = ^Afi. Any measurable subset X of B n may be represented in the form X = ] X k , where k=rn Xk are measurable sets, XkC_A k for fc^n, w+1, ..., and JT/-T/=0 for i^; ; and so $(X)=2$(X k )'^2ak' p(X k )>an-i*(X). We obtain At~ fc^/i thus a descending sequence of measurable sets {B t ,} such that 1 -0) 0(JT)<an-|i(A') if XCE B,,, Xe*, the second relation being obvious, since E B,, is a subset of E A n . Let us now write E l = E B^ E n = B n -\ B n for n = 2, 3, ..., and //^liml?,,. Thus defined the sets H, E^ E 2J ..., E n , ... are mea- ii Ctt surable and without common points, and EH+^E,,. Taking n \ into account the relations (14.5), we see at once that the inequality (14.4) holds whenever X is a measurable subset of E n . Finally, #C#/ for each positive integer n, and therefore, by the first of the rela- tions (14.5), we get $(H)^an /*(//), which requires //(//) () and completes the proof. (14.6) Theorem. If E is a set (9E) of finite measure (//), or, more generally, a set expressible as the s urn of a sequence of sets (9E) of finite measure, every additive function of a measurable set $(X) on E is expressible as the sum of an absolutely continuous additive function W(X) and a singular additive function (X) on E. Such a de- composition of $(X) on E is unique, and the function W(X) ?s, on E, 34 CHAPTER I. The integral in an abstract space. the indefinite integral of a function integrable (9E, /i) on E. If is a non-negative monotone function on E, so are the corresponding functions <P(X) and Q(X). Proof. Since every additive function of a set is the difference of two non-negative functions of the same kind (cf. 6), we may restrict ourselves to the case of a non-negative $(X). Further, we shall assume to begin with that the set E has finite measure. By the preceding lemma, there exists, for every positive integer m, a de- composition of E into a sequence of measurable sets H\ rn \ E\ nt \ E ( "'\ ..., without common points and subject to the conditions: (14.7) E=H (rn) + E\ m) + ... -f E^ + ... , /*(# (m) ) = 0, (14.8) 2~ m .(rc l)./<(A r )<<P(.Y)<2~^n-//(.V), if We therefore have, for all positive integers m, n, and A:, ~~ m 2 and 2 m " 1 k.n(E ( n m} .E ( ^ }) )^ 4>(E ( n m} e^ !) )>2^ (n from which it follows that (2n~ A + l) ^(E^- J5?i //lf !) )>0, and that (fc2n + 2)/4(^r ) -M l "" HI) )^0. Hence ^(BJT - ^ //lfl) )-0 whenever either A:>2w+l, or k<2n 2. We may therefore write (14.9) This being so, let H=JH {m} + V (^ ( ;". We write /<''(a;)-2 '.(n 1) //i, /i- for xeE ( } H, n-1,2,..., and /<"'>(#) = () for (rf//. We thus obtain a sequence </ (m) (^)} of non-negative functions measurable (SE) on the set E. By (14.9) we have clearly |/<"' ( ^(ff) / (m) (^)|<2 ; " on JS, so that the sequence {f (m) (x)} converges uniformly on K to a non -neg- ative measurable function f(x). The set H being of measure zero, we have, by (14.7) and (14.8), for every measurable set X(^E and for every positive, integer m, tf>(X) ^ 4>(X-H) +2 2 - (n 1) -p(X - A 1 ^) = $(X - IT) and [ 14] The Lebesgue decomposition of an additive function. 35 Hence, making w->oo, we derive $(X)= ff(X)du(x)+$(X-H). x This decomposition, so far established subject to the hypo- thesis that the set E is of finite measure, extends at once to sets expressible as the sum of an enumerable infinity of sets of finite measure. In fact, if E=^^A ny where the A n are sets (9E) without n=\ common points and of finite measure, then, by what we have al- ready proved, there exists on A n a non-negative function f n (x) in- tegrable on A n , and a measurable set H n C_A n having measure zero, such that $(X-A n )=ff n dii+0(X'Hn), for n=l, 2, ... If we *4. now write H=]?H n , and /(#)=/(#) for xf A m we obtain a measur- n able set H_E of measure zero, and a function /(Z), non-negative and integrable on E, such that, by Theorem 12.7, for every meas- urable set (14.10) Now, the indefinite integral vanishes for every set of measure zero, and therefore is an absolutely continuous function; on the other hand, we have tf^Z-ffJ^O for every measurable set XC.E H. Thus, since the set H has measure zero, formula (14.10) provides a decomposition of $(X) into an absolutely continuous function and a singular function. Finally, to establish the unicity of such a decomposition, suppose that (X) = & l (X) + 9 l (X) = W 2 (X) + 9 2 (X) on E, the functions ^(Z) and &i(X) being absolutely continuous, and the functions ^(Z) and # 2 (Z) being singular. Then ^(Z) fl r ,(Z) = = $ 2 (Z) ^(Z) identically on J, whence by Theorem 13.1 (2 and 6), we have ^(ZJ-^Z) and ^(ZJ-^Z), and this com- pletes the proof of our theorem. The expression of an additive function as the sum of an ab- solutely continuous function and of a singular function will be termed the Lebesgue decomposition. The singular function that appears in it is often called the function of the singularities of the given function. From Theorem 14.6, we derive at once 36 CHAPTER I. The integral in an abstract space. (14.11) Theorem of Radon- Nikodym. If E is a set of finite measure, or, more generally the sum of a sequence of sets of finite mea- sure (//), then, in order that an additive function of a set (9E) on E be absolutely continuous on E, it is necessary and sufficient that this function of a set be the indefinite integral of some integrable function of a point on E. The hypothesis that the set E is the sum of an at most enumerable infin- ity of sets of finite measure, plays an essential part in the assertion of the theorem of Radon -Nikodym, just as in Theorem 9.8. To see this, let us take again the interval [0, 1] as our space X v and let the class 9^ of all subsets of [0, 1] that are measurable in the Lebesgue sense (cf. below Chap. Ill) be our fixed additive class of sets in the space X. A measure t u l will be defined by taking /^(Z) oo for infinite sets and 4 1 (JT)=n for finite sets with n elements. This being so, the sets (SEj) of measure (/ij) zero coincide with the empty set, and therefore, every additive function of a set (9^) on X l is absolutely continuous (9^, ^). In particular,, denoting by +\(X) the Lebesgue measure for every set X e 9E lf we see that A(X) is absolutely continuous (9^, ft t ) on X v We shall show that A(X) is not an inde- finite integral (9E lf t i<i) on T r Suppose indeed, if possible, that i = J g(x)d t u l (x) x for every set X e 9^, the function g(x) being integrable (9E lf t^) on X r Since A(X) is non-negative, we may suppose that g(x) is so too. Let E= and E,,='E[g(x) -.!/] for n=l, 2, ... We have A(-\\ E)~ fgdft^O. so that xt-E A(E) = A(X l ) = l and this requires the set E to be non -enumerable. Since E=2Eu t n the same must be true of En^ for some positive integer n . Thus n ,) =- fg(x)dfi l (x) ^ ?! (J0 )/n =oo, which is evidently a contradiction. 15. Change of measure. Any non-negative additive func- tion v(X) of a set (SE) may clearly be regarded as a measure cor- responding to the given additive class SE. When such a function v(X) is defined only on a set E, we can always continue it (cf. 5, p. 9) on to the whole space. The terms measure (*>), integral (SE, v) etc. are then completely determined for all sets (9E), but, in this case, it is most natural to consider only the subsets of E for which the function r(X) was originally given. [ 15] Change of measure. 37 (15.1) Theorem. Whenever, on a set E measurable (9E), we have (15.2) v(X) = (90 fg(x)du(x) + (X) x where d(X) is a non-negative function, additive and singular (9E, M) on E, and where g(x) is a non-negative function integrable (9E, p) over E, then also (15.3) (S) fl(x)dv(x) = (S) ff(x)g(x)du(x) + (90 ff(x)d(x) XX X for every set XdE measurable (3t) and for every function f(x) that possesses a definite integral (9E, v) overX. If, further, the function f(x) is integrable (SE, r) over E, the formula (15.3) expresses the Lebesgue decomposition of the indefinite integral f fdv on E, corresponding to the measure p, x the junction 6(X) = / / d9 being the function of singularities (9E, IA) x of the indefinite integral j f dr. x Proof. We may clearly assume that f(x) is defined and non- negative on the whole of the set E. We see at once that, for each set YC% measurable (SE), fc Y (x)dv(x) =v(Y)= fg(x)d t u(x) + d(Y) = Y Y = Jc Y (oo)g(x)dfi(x) + I c Y (x)d9(x), and hence also that for every Y Y finite function h(x) simple and measurable (9E) on a set XC_E, (15.4) fh(x)dv(x) = fh(x)g(x)du(x) + f h(x)d$(x). XX X Let now (h n (x)} be a non-decreasing sequence of finite simple functions measurable .(9E) and non-negative on X, converging to the given function f(x). Substituting h n (x) for h(x) in (15.4) and making n->oo, we obtain (15.3), on account of Lebesgue's Theorem 12.6. If, further, f(x) is integrable (9E, v) over E, the identity just es- tablished shows at once that the product f(x) g(x) is integrable (9E, /*) over E and hence, that the indefinite integral f fg dp is absolutely X continuous (,/*) 'on E. On the other hand, the function 6(X) vanishes on every set on which the function 9(X) vanishes, and therefore, is singular ($,/*) on E together with 9(X). This completes the proof. 38 CHAPTER I. The integral in an abstract space. The wide scope of Theorem 15.1 is due to the fact that, if n(X) and v(X] are any two measures associated with the same class 9E of measurable sets and we have at the same time /*( E)< + > and v(E) < + oo, for a set JEeSE, then the measure v can be repre- sented on E in the form (15.2), where g(x) is a function integrable (9E,^) over the set E and &(X) is a non-negative function, additive and singular (9E, n) on the same set (cf. Th. 14.6). Hence, with the above hypotheses and notation, in order that jfdv= I fgdp x x should hold identically on JE7, it is necessary and sufficient that the indefinite integral ffdv be absolutely continuous (96, p) on E. x This condition is clearly satisfied whenever the measure v(X) is itself absolutely continuous (9f, /*). CHAPTER II. Carath6odory measure. 1. Preliminary remarks. In the preceding chapter, we supposed given a priori a certain class of sets, together with a meas- ure defined for the sets of this class. A different procedure is usually adopted in theories dealing with special measures. We then begin by determining, as an outer measure, a non-negative function of a set, defined for all sets of the space considered, and it is only a posteriori that we determine a class of measurable sets for which the given outer measure is additive. An abstract form of these theories, possessing both beauty and generality, is due to C. Carath^odory [I]. The account that we give of it in this chapter, is based on that of H. Hahn [I, Chap. VI], in which the results of Carath^odory are formulated for arb- itrary metrical spaces. This account will be preceded by two describing the notions that are fundamental in general metrical spaces. 2. Metrical space. A space M is metrical if to each pair a and b of its points there corresponds a non-negative number p(a, ft), called distance of the points a and fc, that satisfies the following conditions: (i) (>(a, 6) = is equivalent to a 6, (ii) p(a, 6) = (>(6, a), (iii) (>(a, b) + Q(b, c)^Q(a, c). In this chapter, we shall suppose that a metrical space M is fixed, and that all sets of points that arise, are located in M. The notation that we shall use, is as follows. A point a is limit of a sequence {a,,} of points in M, and we write a = lima,,, if /i lim(>(a, a,,)=0. Every sequence possessing a limit point is said to 40 CHAPTER II. Carath<k>dory measure. be convergent. Given a set Jf, the upper bound of the numbers p(a, b) subject to aeJMT and beM is called diameter of M and is denoted by d(M). The set M is bounded if rf(Jtf) is finite. For a class 9W of sets, the upper bound of the numbers i(M) subject to JlftSR is denoted by 4(9W) and called characteristic number of 2R. By the distance Q(a, A) of a point a and a set A, we mean the lower bound of the numbers p(a, x) subject to xe A, and by the distance $(A,B) of two sets A and J5, the lower bound of the numbers Q(X, y) for xe A and y e B. We call neighbourhood of a point a with radius r > 0, or open sphere S(a; r) of centre a and radius r, the set of all points x such that (>(a, x)<r. The set of all points x such that Q(a,x)*^r is called closed sphere of centre a and radius r, and is denoted by 8 (a; r). A point a is termed point of accumulation of a set A, if every neighbourhood of a contains infinitely many points of A. The set A' of all points of accumulation of A is termed derived set of A. The set A + A', that we denote by A, is termed closure of ^4. If A = ^.> the set A is said to be closed. The points of a set, other than its points of accumulation, are termed isolated. A set is isolated, if all its points are isolated. We call perfect, any closed set not containing isolated points. A point a of a set A is said to be an internal point of A, if there exists a neighbourhood of a contained in A. The set of all the internal points of a set A is called interior of A and denoted by A. The set A A* is termed boundary of A. If A=-A Q , the set A is said to be open. Two sets A and # are called non-overlapping, if The class of all open sets will be denoted by (& and that of all closed sets by J. In accordance with the convention adopted in 2 of Chap. [, p. 5, open and closed sets will also be termed sets () and sets (ft) respectively. We see at once that the com- plement of any set ((&) is a set (ft) and vice-versa. The sum of a finite number or of an infinity of open sets, as well as the common part of a finite number of such sets, is always an open set. Any common part of a finite number, or of an infinity, of closed sets, and also any sum of a finite number of such sets, [ 2] Metrical space. 41 are closed sets. Nevertheless, the sets (5a) and (<$><>) (cf. Chap. I, 2, p. 5) do not in general coincide with the sets (5) and (), although every set (5) is clearly a set (fta) and, at the same time, a set (<&<>); for, if F is a closed set and G n denotes the set of the points x such that Q(X, F)<l/n, we have F=nO nj where O n are open. The cor- n responding result for the sets (<&) is obtained by passing to the complementary sets. Moreover, it follows that any set expressible as the common part of a set (5) and a set () is both a set (fta) and a set (a). We shall denote by 93, the smallest additive class that includes all closed sets (cf. Chap. I, Th. 4.2). This class, clearly, includes also all sets ((6(5) and (Jo). The sets (93) are also termed measurable (93) (in accordance with Chap. I, 4, p. 7). They are known as Borel sets. We shall also give a few "relative" definitions having re- ference to a set M. The common part of M with any closed set is closed in jtf; we see at once that, for a set PC M to be closed in HI, it is necessary and sufficient that P=M-P, i. e. that the set P contains all its points of accumulation belonging to M . Sim- ilarly, any set expressible as the common part of M and an open set is termed open in M. Any set of the form Jtf-Sfa; r), where aeM and r>0, is called portion of M. If every portion of M contains points of a set A, i. e. if A^) Mj the set A is said to be everywhere dense in M. If a set B is not everywhere dense in any portion of M, i. e. if no portion of M is contained in J5, the set B is said to be non-dense in M. In other words, a set B is non-dense in M , if, and only if, each portion of M contains a portion in which there are no points of B. It fol- lows at once that the sum of a finite number of sets non-dense in the set M is itself non-dense in M. The sets expressible as sums of a finite or enumerably infinite number of sets non-dense in M are termed (according to R. Baire [1]) sets of the first category in Jf, and the sets not so expressible are termed sets of the second category in M. In all these terms, the expression "in Jtf" is omitted when M coincides with the whole space; thus, [by "non-dense sets", we mean sets whose closures contain no sphere and by "sets of the first category", enumerable sums of such sets. A set M is called separable, if it contains an enumerable subset everywhere dense in Af. 42 CHAPTER II. CaratModory measure. 3. Continuous and semi-continuous functions. If f(x) is a function of a point, defined on a set A containing the point a, we shall denote by M^(/; a; r) and m^(/; a; r), respectively, the upper and lower bounds of the values assumed by f(x) on the por- tion A-S(a; r) of the set A. When r tends to 0, these two bounds converge monotonely towards two limits (finite or infinite) which we shall call respectively maximum and minimum of the function f(x) on the set A at the point a, and denote by M^(/; a) and m^(/; a). Their difference o A (f, a) M. 4 (/; a) m^(/; a) will be called oscil- lation of f(x) on A at a. We clearly have (3.1) m^(/; a) ^/(a) ^ M, 4 (/; a) for every point aeA. If f(a) m A (f; a), the function f(x) is said to be lower semi- continuous on the set A at the point a; similarly, if /(a) = M.4(/; a), the function f(x) is upper semi-continuous on A at a. If both con- ditions hold together, and if f(x) is finite at the point a, i. e. if m A (fj a) = M. A (f; a)^ 00 ? the function f(x) is termed continuous on A at the point a. Functions having the appropriate property at all points of the set A, will be termed simply lower semi-continuous, or upper semi-continuous, or continuous, on A. In all these terms and symbols, we usually omit all reference to A, when the latter is an open set (in particular, the whole space), or when A is kept fixed, in which case the omission causes no ambiguity. From these definitions we conclude at once that, if f(x) is upper semi-continuous, the function f(x) is lower semi-continuous, and vice-versa; and further, that, if two functions are upper (or lower) semi-continuous, so is their sum (supposing, of course, that the functions to be added do not assume at any point infinite values of opposite signs). (3.2) Theorem. For every function f(x) defined on a set A, the set of the points of A at which f(x) is not continuous on A, is the com- mon part of the set A with a set (5*). Proof. Let us denote by F n the set of the points x of A at which either f(x) = oo, or o A (f^)^l/n. The set F=2F n con- n sists of all the points of A at which the function f(x) is not con- tinuous. Now it is^easy to see that each of the sets F n is closed in A, i. e. that F n =A-F tl . Therefore F is the common part of A and the set 2F n , which is a set ($). [{3] Continuous and semi -continuous functions. 43 (3.3) Theorem. For a function of a point f(x) to be upper [lower] semi-continuous on a set A , it is necessary and sufficient that, for each number a, the set (3.4) E [x e A ; f(x) > a] [E[xcA; f(x) < a]] X X be closed in A, i. e. expressible as the common part of A with a set (5). Proof. We need only consider the case of upper semi-contin- uous functions, as the other case follows by change of sign. Let f(x) be a function upper semi -continuous on A, a an ar- bitrary number, and x A a point of accumulation of the set (3.4). For each r>0, the sphere S(# ; r) then contains points of that set, and this requires M^(/; x Q j r )^ a an( i 80 M^(/; x Q )^a. Since by hypothesis M^(/; x ) = /(# ), we derive f(x ) ^ a, so that x belongs to the set (3.4). This set is thus closed in A. Suppose, conversely, that the set (3.4) is closed in A for each a. Since the relation M^(/; x) = f(x) is evident for any x at which f(x) -f oo, let x be a point at which f(x ) < + oo, and a any number greater than f(x ). The set (3.4) is closed in A and does not contain # , and so, for a sufficiently small value r n of r, contains no point of the sphere S(# ;r). Thus MA(/; x Q )^M A (f; j? ; r a )^.a for every number a>/(# ), and hence M^(/; # )<:/(# ), which, by (3.1), requires M x (/; x ) = f(x ). An immediate consequence of Theorem 3.3 (cf. Chap. I, 7, particularly p. 13) is the following (3.5) Theorem. Every function semi-continuous on a set (33) is measurable (3J) on this set. More generally, if SE is any additive class of sets including all closed sets (and so all sets measurable (33)), every function semi-continuous on a set (9E) is measurable (9E) on this set. 4. Carath6odory measure. A function of a set defined and non-negative for all sets of the space -Jf , will be called outer measure in the sense of Carathfodory, if it fulfills the fol- lowing conditions: (CJ F(X) ^ F( Y) whenever XCY, (C 2 ) T(S X,)^ 2 P(Xi) for each sequence {X,} of sets, i i (C,) F(X+ Y) - FiX) + 1\ Y) whenever Q(X, Y) > 0. 44 CHAPTER II. Carath4odory measure. It should be noted that of these three conditions, the last one, only, has a metrical character. Now in this , as well as in the 5 and 6, we shall use only properties (d) and (C a ) of the Caratheodory measure. Hence all the re- suits of these remain valid in a perfectly arbitrary abstract space. In order to simplify the wording, we shall suppose, in the rest of this chapter, except in 8 which is concerned with certain Special measures, that an outer Caratheodory measure F(X) is uKiiquely determined in the space considered. , A set E will be termed measurable with respect to the given outer measure F(X), if the relation F(P+Q) = F(P)+F(Q) holds for every paJ^f sets P and Q contained, respectively, in the set E and in its complement CE ^ x , what amounts to the o^^e, if F(X)=F(X.E)+F(X-GE) holds for every set X. By condition (C t ) this last relation may be replaced by the inequality The class of all the sets that are measurable with respect to F 9 will be denoted by fir. We see at once that this class includes all the sets X for which F(X)=Q (in particular, it includes the empty set). Moreover it is clear that complements of sets (fir) are also sets (fir). The main object of this is to establish the additivity of the class fir (in the sense of Chap. I, 4) and to prove that the function F(X) is a measure (fir) in the sense of Chap. I, 9. This result will constitute Theorems 4.1 and 4.5. (4.1) Theorem. If 8 is the sum of a sequence (X n } n =\. 2, ... of sets (fir) no two of which have common points, the set 8 is again a set (fir) and F(S) = 2 r(X n )-j more generally, for each set Q (4.2) F(Q) = 2 F(Q .Z n ) + F(Q C8). n k Proof. Let 8*= 2 X n . We begin by proving inductively that /i i all the sets S k are measurable with respect to 1\ and that, for each k and for every set Q, (4.3) F(Q) > 2 F(Q.X n ) + F(Q.Q8 k ). /|:rrl Suppose indeed, that 8 P is a set (fir) aud that the inequality (4.3) holds for every set Q, when k = p. Since X p +i is, by hypothesis, a set (fir) and S p -X p +i = 0, we then have [4] Carath^odory measure. 45 ) + F(Q.CX p+} ) = i-s,,) + r(Q.cx p+t .cs p ) = n 1 and this is (4.3) for Jc = p-\-l. In view of condition (0 2 ), p. 43, it follows further that F(Q)^ r(Q-8 p + 1) + r(Q-C8 p + } ), which proves that S p +i is a set (fir). Combining the inequality (4.3), thus established, with the in- equality r(Q-CSk)^ r(Q-CS}, we obtain, by making fc->oo, the inequality 1"(Q) ^ j? F(Q. X n ) + r(Q-CS), and from this (4.2) /i- 1 follows on account of condition (C 2 ). Finally, the same condition enables us to derive from (4.2) that r(Q)^r(Q.8) + r(Q.C8), and this shows that 8 is a set (fir) and completes the proof. (4.4) Lemma. The difference of two sets (fir) is itself a set (fir). Proof. Let Jfefir and Ye fir, and let P and Q be any two sets such that PCX Y and #CC(Z ). Write Q l = Q-T and Q 2 = Q - C Y. Making successive use of the three pairs of in- clusions QtCY, # 2 CCr ; PCX, Q 2 CC(X Y).CYCCT; and p+Q 2 ccY, we find r(p)+r(Q)=r(p)+r(Q l )+r(Q t )^ s) + r(Q 1 )=r(P+Q), which shows that X Y is a set (fir). (4.5) Theorem. Zr is an additive class of sets in the space M. Proof. We have already remarked (p. 44) that the empty set and that complements of sets (fir) are sets (fir). To verify the third condition (iii) for additivity (cf. Chap. I, 4, p. 7), let us observe firstly that, on account of Lemma 4.4 and of the identity X-Y=X CY, the common part of any two sets (fir) is itself a set (fir). This result extends by induction to common parts of any finite number of sets (fir) and, with the help of the identity Xi~ CllCXf, we pass to the similar result for finite sums of sets. Finally, if X is the sum of an infinite sequence {X n } n , i. 2t ... of sets (fir), we have X=S l +2 (8,,+i S n ) where 8 n = X*. Now, /i-l /r-l clearly, of the sets 8 l and S n +i $, no two have common points, and, moreover, by the results already proved, they all belong to the class fir. Consequently, to ascertain that X is a set (fir)> we have only to apply Theorem 4.1. The class fir is thus additive. 46 CHAPTER II. Carathdodory measure. Theorem 4.5 connects the considerations of this chapter with those of the preceding one. Thus, in accordance with the conven- tions adopted in Chap. I, pp. 7 and 16, the sets (fir) may be termed sets measurable (fir), and F(X) may, for .Xefir, be regarded as a measure associated with the class fi/\ This class, together with the measure J 1 , determines further the notions of functions measurable (fi/^), of integral (8/> F), of additive function of a set (fir) absolutely continuous (fi r , F), and the other notions defined generally in Chap. I. Since the outer measure F determines already the class fir, we shall omit in the sequel the symbol representing this class, whenever the notation makes expli- cit reference to the outer measure; thus we shall say "f unction integrable (/')" instead of "function integrable (fir, JT)" and the integral (JT) of a function f(x) over a set E will be denoted simply by ff(x)dF(x), instead of by (Z r ) ff(x)dF(x). K E In accordance with Chap. I, 9, the value taken by F(X) for a set X measurable (fir) will be termed measure (F) of X] when X is quite arbitrary, this value will be called its outer measure (T). If JS is a subset of a set E such that F(E E ) =0, then for any function f(x) on E the measurability (fir) of / on E is equivalent to its measurability (fir) on E Q . This remark and Theorem 11.8, Chap. I, justify the following convention: If a function /(#) is defined only almost everywhere (F) on a set E, then, E denoting the set of the points of E at which f(x) is defined, by measurability (fir), integrability (F) and integral (F) of f on the set E we shall mean those on the set J5 . Let us note two further theorems. (4.6) Theorem. Given an arbitrary set E, (i) F(E.2X n )=SF(E.X n ) n n for every sequence {X n } of sets measurable (fir) no two of which have common points, (ii) F(E- limX,,) = limF(E-X n ) for every ascending n n sequence {X,,} of sets measurable (fir), and this relation remains valid for descending sequences provided, however, that F(E-X l )=^oo, (iii) more generally, for every sequence {X n } of sets measurable (fir) F(E-limiitiXn)^limiDtr(E-X H ), and, if further F(E-2X n ) ^=00, n ii n then also P(E- lim sup X n ) ^ lim sup F(E- X n ). I 5] The operation (A). 47 Part (i) of this theorem is contained in Theorem 4.1, and parts (ii) an(J (iii) follow easily from (i) (cf. Chap. I, the proofs of Theorems 5.1 and 9.1). A part of Theorem 4. 6 will be slightly further generalized. Given a set E, let AIS denote, for any set X, by / ' (X ) the lower bound of the values taken by F(E-Y) for the sets Y measurable (2/') that contain X. (4.7) Theorem. Given a set E, (i) to every set X corresponds a set X Q ^)X measurable (2r) such that r% (X) = F(E- Z), (ii) r(J57.1iminfXiXJ r ^(liminf^ /1 )^liminf/ 1 ? ; (Z, l ) for every se- ii n 11 quence {X n } of sets, and, in particular, F(E - lim X lt ) ^ /^(lim X u ) it n lim ri(X n ) for every ascending sequence (X,,}. n Proof, re (i). For every positive integer n there is a set Y n D JT, measurable (fir), such that P(E- YJ^-TJ^JO + l/n. Writing X Q fJ Y,,, we verify at once that the set X Q has the required /i properties. re (ii). Taking (i) into account, let us associate with each set X n a set XnDX,,, measurable (2r) and such that r(E-Xl) = Fl(X n ). The set liminf JT^ 3) liminf X n is measurable (2j') and, we therefore have, by Theorem 4. 6 (iii) PS (liminf X n ) ^ f(E liminf X) ^ liminf r(E-X^)-. n n n n The second part of (ii) follows at once from the first part. * 5. The operation (A). We shall establish here that measurability (2/0 is an invariant of a more general operation than those of addition and multiplication of sets. We call determining system, any class of sets ^l={A nitn ., t ,,, tflh } in which with each finite sequence of positive integers n^n 2 , ...,n/, there is associated a set -A,,,,,,, , J/r The set where the summation extends over all infinite sequences of indices n l} n 2 , ..., nk, ..., is called nucleus of the determining system W and denoted by N($l). The operation leading from a determining system to its nucleus is often called the operation (A). 48 CHAPTER II. Carath^odory measure. The operation (A) was first defined by M. Souslin [1] in 1917. When applied to Borel sets, it leads to a wide class of sets (following N. Lusin, we call them analytic) and these play an important part in the theory of sets, in the theory of real functions, and even in some problems of classical type. A system- atic account of the theory of these sets will be found in the treatises of H. Hahn [II], F. Hausdorff [II], C. Kuratowski [I], N. Lusin [II] and W. Sierpin- ski [II]. We* mentioned at the beginning of this J that the operation (A) includes those of addition and of multiplication of sets. This remark must be understood as follows: IfWlisa class of sets such that the nucleus of every determining system formed of sets (3K) itself belongs to 3H, then the sum and the common part of every sequence {Nt} of sets (3R) are also sets(W). In fact, writing P/, 1 ,/i 2 ,...,/ A =^ r /i 1 and Qn^n^. .,n k =Nk we see at once that the nuclei of the determining systems {P// 1 ,/i 2 ,...,n^} and {Q/ij,/i 2 . . ,n*} coincide respectively with the sum and with the common part of the sequence {Ni}. Thus, Theorem 5,5, now to be proved, will complete the result contained in Theorem 4.5, and in conjunction with Theorem 7.4, establish mea- surability (C/0 for analytic sets in any metrical space (cf. N. Lusin and W. Sierpinski [1], N. Lusin [3, pp. 2526], and W. Sierpinski [12; 15]). The proof of this can be simplified if we assume regularity of the outer meas- ure 7" (cf. 6) (see C. Kuratowski [I, p. 58]). With every determining system 91 (A nit n^... t n k } 9 we shall also associate the following sets. N*i*2' * (91) will denote, for each finite sequence h l9 A 2 , ..., h y of positive integers, the sum (5.1) extended over all sequences, w 1? n 2 , ..., n*, ... such that n/^A/ for 1=1,2, ...,*. We see at once that the sequence {1^(21)^=1,2,..., together with every sequence {N*i*2 ..... ** A (2l)}fc=i f 2,..., is monotone ascending and that (5.2) N(9l) = limN*(9l), M^-^ ..... **(2i) = limN*^. -.**. *(). h h Further, for every sequence of positive integers h^ ft a , ..., A*, ..., we shall write n, i /i, = /, - We see directly that if the sets of the determining system 21 belong to a class of sets 901, the sets N Al ,/, 2 ..... h k (^i) belong to the class [ 5] The operation (A). 49 (5.3) Lemma. For every determining system yi={A ni , n2 ..... llft ) and for every sequence of positive integers h^ ft 2r ..., ft*, ... (5.4) N Al (9l) N^Sl). ....N,,^,...,^) - ...C N(9i), Proof. Let a? be any point belonging to the left-hand side of (5.4). We shall show firstly that a positive integer n^^^ can be chosen so that, for each fc^2, the point x belongs to a set A ni -An lt n 2 '..-'A ni ,n 2 ..... n k for which tt^tt? and n/<ft/ for i=2, 3, ..., fc. Indeed, if there were no such integer nj, we could associate with each index n^h lr a positive integer &, such that x belongs to no product An'A ntn2 -...-A nt n 2 ,...,n k for which n^hi when i=2,3, ..., k n . Denote by p l the greatest of the numbers fcj, Jc 2 , ..., fc A| . The point x thus belongs to none of the sets A /ll -A /I1 , / , 2 -...-^l / , 1 ,, l2 ,..., n for which ni^hi when i = 1, 2, ..., p x , and therefore is not contained in their sum NA^ ..... /, (91). This is a contradiction since, by hypo- thesis, x is an element of the left-hand side of (5.4). After the index nj, we can determine afresh an index so that, for each fc^3, the point x belongs to a set A,,^,,^-...-^,,^,...,,^ for which n v = n, n% = n? 2 and n^hi when i = 3, 4, ..., fc. For, if there were no such index, we could find, as previously, a positive integer p^3 such that x belongs to no product A n} >A ni , n2 '...-A nit n 2 ..... rip for which n l =n Q l and n^hi when i=2, 3, ..., p 2 - And this would contradict the definition of the index n Q { . Proceeding in this way, we determine an infinite sequence of indices {n} such that rf^hi when i = 1, 2, ... and such that #e jL n o.J. n o >n p.....J. n oy>^ y>.... . Thus #eN(21), and this completes the proof. Lemma 5.3 is due to W. Sierpinski [13]. The proof contains a slightly more precise result than is expressed by the relation (5.4) and shows that the left-hand side of that relation coincides with the sum (5.1), when the latter is extended only to systems of indices n lt n 2 , ..., nk, ... restricted to satisfy tij ^ 7ij, n-2 <? /&2 > n k ^ ^A' ** * Let us call degenerate, a determining system {An^.^...^^} such that, for some sequence {hk} of positive integers, we have ^/ij,/i 2 , ...,^ whenever nk - hk< Then, for this sequence {hk}, the relation of inclusion (5.4) becomes an identity and we are led to the following theorem: If a degenerate determining system consists of sets belonging to a class 'JR, its nucleus is a set (Wldafi). A similar theorem cannot hold for non -degenerate systems: in fact, as shown by M. Souslin, the operation (A) applied to Borel sets (and even to linear segments) may lead to sets that are not Borel sets (cf. F. Hausdorff [II, p. 182184]). 50 CHAPTER II. Carath^odory measure. (5.5) Theorem. The nucleus of any determining system < yi=\A nitn consisting of sets measurable (fir) is itself measurable (fir). Proof. Let us write, for short, N = N(9l), tfi."2.-.* = Ni.2. .."* (), #,,,....., = N,,^,.. We have to show that, for any set E (5.6) r(E)^F(E'N) + r(E-GN). We may assume that T(?) < oo, since (5.6) is evidently fulfilled in the opposite case. Let us denote (as in 4, p. 47) by r%(X) the lower bound of the values of F(E-Y) for sets T^)X measurable (fir) and let e be an arbitrary positive number. Taking into account (5.2) and Theorem 4.7, we readily define by induction a sequence of positive integers {h k } such that r$(N**)^ r(E-N) e/2 and *'-'-' h *-i) cl2* for fc = 2, 3, ... . Thus the sets N ni , n2t ..^D^" 1 '" 2 ' '"* being measurable (fi/0 together with the An t n t ...,n 9 for each fc, and therefore ' Now the sequence of sets {-N r /, 1 ,/, 2 ,...,/, A }/ f -i,2, .. is descending, and byLemma5.3 its limitis a subset of N. The sequence {0^^...,^}*- ,1,2,... is thus ascending and its limit contains the set CN. Hence, making &->oo in (5.7), we find, by Theorem 4.6(ii), the inequality r(E)^r(E.N)+r(E-CN) e, and this implies (5.6) since f is an arbitrary positive number. 6. Regular sets* A set X will be called regular (with respect to the outer measure JT), if there exists a set A measurable (C/')> containing X and such that r(A) J J (X). Every measurable set is evidently regular, and so is also every set X whose outer measure ( T) is infinite, since we then have r(X)=r(M)=oo. If every set of the space considered is regular with respect to the outer measure JT, this measure is itself called regular; cf. H. Hahn [I, p. 432], C. Carath^odory [1; I, p. 258]. [ 7] Bore! sets. 51 Denoting by r*(X) the lower bound of the values of for sets T^)X measurable (fir), we see readily that the relation r(X) = P(X) expresses a necessary and sufficient condition for the set X to be regular. From Theorem 4.7(ii), taking for the set E the whole space, we derive the following: (6.1) Theorem. For any sequence {X n } of regular sets JT(limmf X n )^. n ^ liminf F(X n ) 9 and, if further the sequence {X,,} is ascending The generality and the importance of this theorem consist in that all outer measures F that occur in applications satisfy the condition of regularity, and, for these measures, the last relation of Theorem 6.1 therefore holds for every v ascending sequence of sets. Nevertheless, for measures that are not them- selves regular, the restriction concerning regularity of the sets X n is essential for the validity of Theorem 6.1 as is shown by an example of irregular measure due to C. Carath^odory [II, pp. 693 696]. We may observe further that, for any fixed set E* the function of a set / #( JT), defined in 4, p. 47, is always a regular outer measure, even if the given measure P(X) is not. Conditions (C 1 ) and (C 2 ) together with that of regularity, are at once seen to hold, and (C 8 ) may be derived from Theorem 7.4, according to which closed sets are measurable 7, Borel sets. We shall show in this that, independently of the choice of the outer measure i 7 , the class C/' contains all Borel sets. (7.1) Lemma. If Q is any set contained in an open set G, and Q n denotes the set of the points a of Q for which $(a, CG)^l/n, then lim r(Q n ) - F(Q). n Proof. Since the sequence {Q n } is ascending and Q = lim Q nj n it suffices to show that lim r(Q n )^r(Q). For this purpose let us write D n =Qn+iQn. We" then have Q(D n +\, <M^l/n(n+l)> 0, provided that D,,+i=f=0 and <?/,4=0. Hence, taking into account condition (C 3 ), p. 43, it is readily verified by induction that (7.2) r(# 2n +o>AJ;i>2A)=J^ t) for every positive integer n. Writing, for short, a fl = r(D^) and k--n 52 CHAPTER II. Carath^odory measure. b n = r(D 2 *_i ), we obtain at once, by condition (C t ), p. 43, (7.3) " F(Q) < r(Qi n ) + a n + b n . OO Now two possibilities arise: either both series. 5/ F(D^) and A-l 00 T F(Du-i) have finite sums, or, one at least has its sum infinite. /& In the former case a,,->0 and fr/,->0, so that, the required inequal- ity F(Q)^limF(Q n ) follows by making n-+oo in (7.3); while, in /i the latter case, the inequality is obvious, since by (7.2) we have then lim F(Q a ) = oo. n (7.4) Theorem. Every set measurable ( S B) is measurable (C r ). Proof. Since the class 2r is additive and since 93 is the smallest additive class including the closed sets (cf. 2, p. 41), it is enough to prove that every closed set is measurable (2/0? i- ?> denoting any such a set by F, that (7.5) r(P+Q)^F(P) + F(Q) holds for every pair of sets PC.& and Q(ZCF. Since the set CF is open, there is, by Lemma 7.1, a sequence {Q n } of sets such that QnCQ, 9(Q, F) ^ l/n for n-1, 2, ..., and limF(Q n ) = F(Q). Thus P) ^ Q(Qni F) > 0, and so, on account of condition (C 3 ), p. 43, we derive F(P + Q)^F(P + Q n ) = F(P) + F(Q n ) for each n, and, making n->oo, we obtain (7.5). The arguments of this depend essentially on property (C 3 ) of outer meas- ure, and on the metrical character of the space 37, which did not enter into 4 6. It is possible however to give to these arguments a form, independent of condition (C 3 ), valid for certain topological spaces that are not necessarily metrical (cf. N. Bourbaki [1]). From the preceding theorem coupled with Theorem 3.5, we derive at once the following (7.6) Theorem, (i) Every function measurable (33) on a set E is measurable (2/ ) on E. (ii) Every function that is semi-continuous on a set (2r) is measurable (2/-) on this set. [8] Length of a get. 53 8. Length of a set. We shall define in this a class of functions of a set that are outer Carath^odory measures and that play an important part in a number of applications. Let a be an arbitrary positive number. Given a set X , we shall denote, for each e>0, by A ( (X) the lower bound of the oo sums ,[<*( -/)]", for which {-Z/}/=i,2,... is an arbitrary partition of X into a sequence of sets that have diameters less than e. When >Q, the number A^(X) tends, in a monotone non-decreasing manner, to a unique limit (finite or infinite) which we shall denote by A a (X). The function of a set A a (X) thus defined is an outer measure in the sense of Carath^odory. For, when OO, we clearly have (i)A2(X)^A ( ?(Y)itXCY, (ii) A?(2x n )^ZA ( *\X n ), X{X n } n n is any sequence of sets, and (iii) A ( *\X+ Y) = A ( ( ?(X) + ^(Y), if $(X, Y)>f. Making f->0, (i), (ii), (iii) become respectively the three conditions (t^), (C 2 ), (C 3 ), p. 43, of Carath^odory for A n (X ). We shall prove further that the outer measure A a (for any a>0) is regular in the sense of 6, i. e. that every set is regular with respect to this measure. We shall even establish a more precise result, namely (8.1) Theorem. For each set X there is a set He* such that XCH and A ft (H) = A a (X). Proof. For each positive integer w, there is a partition of X into a sequence of sets {X^ n) }i 1,2, .. such that (8.2) (JU; 7l) )<l/2n for t=l,2,..., and ?[d(jfi)y^A t (X) + lln. i- \ We can evidently enclose each set X^ in an open set G ( " } such that (8.3) 00 00 Writing // = /"/ 0,, we see at once that H is a set ((&,>) and that /i i i i XCH. Moreover, for each n, H = H-Q\ n} and the relations (8.2) / i and (8.3) imply that 6(H-G\ H} )<lln for =1, 2, ... and that A ( l n \H) ^ Z[6(H- 0j n) )] < (1 +l/w)" [ A (( (X)+l/n]. Making n -> oo, t-\ we find in the limit A l (H)^A a (X), and, since the converse in- equality is obvious, this completes the proof. 54 CHAPTER II. Carath^odory measure. In Euclidean n-dimensional space R n (see Chap. Ill), the sets whose measure (A n ) is zero may be identified with those of measure zero in the Lebesgue sense. By analogy, in any metrical space, sets whose measure (A a ) is zero are termed sets having a-dimensional volume zero, and in particular, when a = l, 2, 3, sets of zero length, of zero area, of zero volume, respectively. For the same reason, sets of finite measure (A,?) are termed sets of finite a-dimensional volume (or of finite length, finite area, finite volume, in the cases =1, 2, 3). In particular, in H 19 i. e. on the straight line, the outer measure A l coincides with the Lebesgue measure, and, on this account, we call the number A^X), in general, outer length of X, and when X is a set measurable (2 ,), simply, length of X. For short, we often write A instead of A v We have mentioned only the more elementary properties of the measures A tt , those, namely, that we shall have some further occasion to use. For a deeper study, the reader should consult F. Hausdorff [1]. Among the researches de- voted to the notion of length of sets in Euclidean spaces, special mention must be made of the important memoir of A. S. Besicovitch [1]; cf. also W. Sier- pinski [I] and J. (rillis [1], 9. Complete space. A metrical space is termed complete, if a sequence {a n } of its points converges whenever lim Q(a m , a ;i )=0. //I,fl->OO In any metrical space, this is evidently a necessary condition for convergence of the sequence {a,,}, but, as a rule, not a sufficient one. The following theorem concerns a characteristic property of complete spaces: (9.1) Theorem. In a complete space, when {F fl } is a descending sequence of closed and non-empty sets whose diameters tend to zero, the common part IIF n is not empty. Proof. Let a n be an arbitrarily chosen point of F n . For we have Q(OJ,,, a tl )^d(F m ), and hence lim e(a,,,, a/,)=0. The sequence {a,,! is thus convergent. Now the limit point of this sequence clearly belongs to all the sets F inj since a n F n C.F m whenever n^m, and since the sets F n , are closed by hypothesis. (9.2) Baire's theorem. In a complete space M, every non-empty set (<&,}) M of the second category on itself, i. e. if H is a set (<j) in M and H=S H n , one at least of the sets H n is everywhere dense n * in a portion of U. [9] Complete space. 55 oo Proof. Suppose, accordingly, that H = H G n where #/, are open sets, and further that / f\ O\ TT V TT ("<3) J = 2j *ln n=^l where H n are non-dense in H. The partial sums of the series (9.3) are then also non-dense in H (cf. 2, p. 41), and it is easy to define inductively a descending sequence of portions S n of H n such that (i) S a CQn, (ii) Sn- #p=0, (iii) 6(8 n )^ln. On account of Theorem 9.1 and of (iii), the sets S n have a common point, which by (i), belongs to all the sets (?, and so to H, while at the same time, by (ii) it belongs to none of the H n . This contradicts (9.3) and proves the theorem. The case of Theorem 9.2 that occurs most frequently, is that in which JET is a closed set. For closed sets in Euclidean spaces the theorem was established in 1899 by R. Baire [1], To Baire, we owe also the fundamental applications of the theorem, which have brought out the fruitfulness and the importance of the result for modern real function theory. As regards the theorem by itself however, it was found, almost at the same time and independently by W. F. Osgood [1] in connection with some problems concerning functions of a complex variable (cf. in this connection, the interesting article by W. H. Young [7]). The general form of Theorem 9.2 is due to F. Hausdorff [I, pp. 326 328; II, pp. 138145]. If a is a non-isolated point (cf. 2, p. 40) of a set M , the set (a) consisting of the single point a is clearly non-dense in M . It there- fore follows from Theorem 9.2 that (9.4) Theorem. In a complete metrical space, every non-empty set ((6<j) without isolated points, and in particular every perfect set, is non-enumerable. More precisely, by a theorem of W. H. Young [I], every set that fulfills the condition of Theorem 9.4 has the power of the continuum; cf. also F. Haus- dorff [II, p. 136]. CHAPTER III. Functions of bounded variation and the Lebesgue-Stieltjes integral* g 1, Euclidean spaces. In this chapter, the notions of measure that we consider undergo a further specialization. Accord- ingly we introduce for Euclidean spaces, a particular class of outer measures of Carath^odory, determined in a natural way by non-negative additive functions of an interval. These outer meas- ures in their turn determine the corresponding classes of meas- urable sets and measurable functions, and lead to processes of integration usually known as those of Lebesgue-Stieltjes. By Euclidean space o/ m dimensions lt mj we moan the set of all systems of m real numbers (#1, j? 2 , ..., #m). The number x h is termed fc-th coordinate of the point (j?i, j? 2 j > ^/n) The point (0, 0, ..., 0) will be denoted by 0. By distance Q($,y) of two points x^~(x\, x, ...,,/:,) and y--=(y\jy>j ...,/////) in the space U mj we mean the non-negative number [(//, j'i) 2 + 0/2 i) 2 f 4- (?//// #/;J 2 | 1j - Distance, thus defined, evidently fulfills the three conditions of ('hap. II, p. 40, and hence Euclidean spaces may be regarded as metrical spaces. All the defi- nitions adopted in ('hap. II therefore apply in particular to spaces JR m . In 2 we supplement them by some definitions more exclusively restricted to Euclidean spaces. The space K l is also termed straight line and the space /? 2 , plane. Accordingly, the sets in /? t will often be called linear, and those in /? 2 plane stts. [ 2] Intervals and figures. 57 $ 2. Intervals and figures. Suppose given a Euclidean space B m . The set of the points (xi, a? 2 , -, x m ) of R m that fulfill a linear equation a\x\+aiXi+ ...+a m x m ~b, where 6, ai, a 2 , ..., a. n are real numbers and, of these, the coefficients a b a 2 , ..., a,/, do not all vanish together, is called hyperplane aiXi+aiXi+...-\-a m x m =b. For each fixed fc=l, 2, ..., m, the hyperplanes a?*=6 are said to be orthogonal to the axis of x k . The term hyperplane by itself, will be applied ex- clusively to a hyperplane orthogonal to one of the axes. In IZ lf hyperplanes coincide with points. In JB 2 and J? 3 they are respectively straight lines and planes. Given two points a=(aj, a 2 , ..., a m ) and 6==(6i, 62, ..., 6 m ) such that a k ^.b k for A=l, 2, ... , m, we term closed interval [01>6|; 62)62; ...; <ljnj 6m] *ke Set * a ^ 'ke P ^ 8 (#1, #2, , J?m) SUCh that 64 ^J?*^ 6* for i=l, 2, ..., m. The points a and b are called principal vertices of this interval. If, in the definition of closed in- terval, we replace successively the inequality a* <I x k ^ b k by the in- equalities (1) 6ft<#ft<6ft, (2) 0ftOft<6ft and (3) a k <x k ^b k , we obtain the definitions (1) of open interval (a\ 9 b\\ a^ 6 2 ; ; #///> bm)> (2) of interval half open to the right [i, ftij ^2, 62; ; /n> 6//i) and (3) of interval half open to the left (a\ 9 b\; a>, 62; ; ^/n> b m ]. If ak=bk for at least one index A:, all these intervals are said to be degenerate. In what follows, an interval, by itself, always means either a closed non-degenerate interval or an empty set, unless another meaning is obvious from the context. We call face of the closed interval I=[ai, b\\ 02, 62; ; #/, b m ] the common part of / and any one of the 2m hyperplanes .*=#* and Xk=btt, where X'=l, 2, ..., m. If J is an open or half open in- terval we call faces of J those of its closure J. We see at once that the faces of any non-empty interval / are degenerate intervals and that their sum is the boundary of the interval 7. If 61 di-&2~ 02=...=6 OT a m ^0, the interval [ai,&i; 02,62;..-; /, 6 m ] is termed cube (square in JK 2 ). We define similarly open cubes and half open cubes (half open to the right or to the left). We call net of closed intervals in H m any system of closed non- overlapping intervals that together cover the space M m . Similarly, by net of half open intervals, we mean a system of intervals half open on the same side, no two of which have common points, and whose sum covers JK m . A sequence of nets {9U} (of closed or of half open intervals) is regular, if each interval of 9U+1 is contained in 58 CHAPTER III. Functions of bounded variation. an interval of 9U and if the characteristic numbers 4(9U) (cf. Chap. II, p. 40) tend to as &->oo. Given a net of half open inter- vals, we clearly change it into a net of closed intervals by replacing the half open intervals by their closures. The same operation changes any regular sequence of nets of half open intervals into a regular sequence of nets of closed intervals. (2.1) Theorem. Given an enumerable system of hyperplanes x k =aj, where k 1, 2, ..., w, and j = l, 2, ..., we can always construct a regular sequence {91*} of nets of cubes (closed or half open) none of which has a face on the given hyperplanes. To see this, let b denote a positive number not of the form qaj/p where p and q are integers and j=l, 2, .... Such a number b certainly exists, since the set of the numbers of the form qaj/p is at most enumerable. This being so, for each positive integer k let us denote by 9U the net consisting of all the cubes half open to the left ( Pl 6/2*, ( Pl + 1)6/2*; p 2 6/2*, (p 2 + 1)6/2*; ...; P/n 6/2*, (p m + 1)6/2*], where p\, p 2j ..., p m are arbitrary integers. The sequence of nets {yik} evidently fulfills the required conditions. Let us observe that, given a regular sequence {9U} of nets of half open [closed] intervals, every open set G is expressible as the sum of an enumerable system of intervals (9U) without common points [non-overlapping]. To see this, let 2)^ be the set of intervals of 9^ that lie in (?, and let 9JU+i, f r eac h *^1? be the set of intervals (9l*+i) that lie in G but not in any of the intervals (3JU). Since 4(9fo)->0 as k -> oo, the enumerable system of intervals ]? 9DU covers the set (?, and the other conditions required are evidently satisfied also. On account of Theorem 2.1, we derive at once the following proposition which will often be useful to us in the course of this Chapter: (2.2) Theorem. Given a, sequence of hyperplanes {#/}, every open, set G is expressible as the sum of a sequence of half open cubes without common points [or of closed non-overlapping cubes] whose faces do not lie on any of the hyperplanes Hi. A set expressible as the sum of a finite number of intervals will be termed elementary figure, or simply, figure. Every sum of a finite number of figures is itself a figure, but this is not in general the case for the common part, or for the difference, of two figures. [ 3] Functions of an interval. 59 We shall therefore define two operations similar to those of multi- plication and subtraction of sets, but which differ from the latter in that, when we perform them on figures, the result is again a figure. These operations will be denoted by and and are defined by the relations AQB=(A^BY and AQB = (A^~BT. The relation AQB = means that the figures A and B do not overlap (cf. Chap. II, p. 40). Given an interval /=[ai, ftj; a>, b^ ...; a m , 6 m ], the number (bi a\)'(b ai)-. (b m a m ) will be called volume of / (length for m=lj area for w = 2), and denoted by L(/) or by |/|. If 7=0, by L(/)=|/| we mean also. When several spaces ft m are considered simultaneously, we shall, to prevent any ambiguity, denote the volume of an interval 7 in lt m by L m (/). We see at once that every figure R can be subdivided into a finite number of non-overlapping intervals. The sum of the volumes of these intervals is independent of the way in which we make this subdivision; it is termed volume (length, area) of R and denoted, just as in the case of intervals, by L(#) or by \R\. 3. Functions of an interval. We shall say that F(I) is a function of an interval on a figure R [or in an open set 0], if F(l) is a finite real number uniquely defined for each interval / contained in R [or in G\. To simplify the wording, we shall usually suppose that functions of an interval are defined in the whole space. A function of an interval F(I) will be said to be continuous on a figure R, if to each f>0 there corresponds an //>0 such that |/|<T? implies \F(I)\<c for every interval IdJR. A function of an interval will be said to be continuous in an open set G, if it is continuous on every figure RC_G. Finally, functions continuous in the whole space will simply be said to be continuous. The reader will have noticed that we use the terms "on" (or "over") and "in" in slightly different senses. We may express the distinction as follows. Suppose that a certain property (P) of func- tions of a point, of an interval, or of a set has been defined on fig- ures. We then say that a function has this property in an open set G, if it has the property on every figure RC_G. Further, if a function has the property (P) in the whole space, we say simply that it has the property (P). Thus, for instance, a function 60 CHAPTER III. Functions of bounded variation. of a set Q(X) is additive (33) (cf. Chap. I, 5 and Chap. II, 2) in an open set G, if it is additive (33) on every figure JRC#; if ^ is a measure (33) (cf. Chap. I, 9), a function of a point, de- fined in the whole space, is daid to be integrable (33, //), if it is integrable (33, fO on every figure, and so on. We shall call oscillation Q(F; E) of a function of an interval F on a set E, the upper bound of the values \F(I)\ for intervals IE. If D is an arbitrary set and R a figure, we shall denote by o/?(F;D) the lower bound of the numbers O(F; R-G), where G is any open set containing Z); the number o R (F;D) will be termed oscillation of F on R at the set D. Finally we shall say simply oscillation of F at Dj and use the notation o(F; D), for the upper bound of the numbers o/i(F; />) where R denotes any figure (or, what amounts to the same, interval or cube). In the sequel, D will usually be a hyperplane (a point in K 19 a straight line in /? 2 ) or else the boundary of a figure. In these cases we shall say that the function F is continuous, or discontinuous, at D on R, according as Ojt(F; Z>)=0 or o/ ? (F;/>)>0. Similarly, we shall say that F is continuous, or discontinuous, at D, according as o(F-,D) = or o(F;D)>0. (3.1) Theorem. In order that a function F of an interval be con- tinuous on a figure R [or in tfa whole space], it is necessary and suf- ficient that o/ ? (P;Z)) = [or that o(F;D) = 0] for every hyper- plane D. Proof. Since the condition is clearly necessary, let us suppose that the function F is not continuous on R. There is then a number f >0 and a sequence of intervals {I (n} ^[a\"\ M' ; ...; a%\ b^]} contained in R and such that for n=l, 2, ..., \F(l (n) )\ >* , and that |7 (ll) |-0. By the second of these conditions, we can extract from the sequence {l (n) }, a subsequence {/ }, in such a manner that lim (b* k a k )^() for a positive integer i i ^m. The sequences Wo }*-i 2 au( * 'fa* 9* 12 fc ^ en have a common limit point a, and, denoting by D the hyperplane #/ o =a, we see that every open set <O/> contains an infinity of intervals / (//) , so that l41 Functions of bounded variation. i 4. Functions of an interval that are additive and of bounded variation. A function of an interval F(I) is said to be additive on a figure R [or in an open set 6?], if F(I l +I 2 )^ =F(I l )+F(I 2 ) whenever I 19 1 2 and /i+/ 2 are intervals contained in R [or in G] and I 19 I 2 are non-overlapping. A function additive in the whole space, is for instance the volume L(/)=|/|. Just as in the case of the function L(I) (cf. 2, p. 59), every additive function of an interval F(I) on a figure R [or in an open set 0] can be continued on all figures in R Q [or in O] in such a manner that F(R l +R 2 )=F(R 1 )+F(R 2 ) for every pair of figures /^C-Ro an(i ^aC-Ro that do not overlap. In the sequel, we shall always sup- pose every additive function of an interval continued in this way on the figures. If F is an additive function of an interval on a figure /? , we shall term respectively upper and lower (relative) variations of F on R the upper and lower bounds of F(R) for figures R^R^. We denote these variations by W(F: R ) and W(F; R ) respectively. Since every additive function vanishes on the empty set, we have W(JF; JR )>O^W(F;Jf? ). The number W(F;7? )+ W^; J? )|, clearly non-negative, will be called absolute variation of F on R Q and de- noted by W(JP; R ). If W(jP; # )<+oo, the function F is said to be of bounded variation on R Q . In accordance with the convention of 3, p. 59, an additive function of an interval in the whole space is of bounded variation, if it is of bounded variation on every figure. It is obvious that a function of bounded variation on a figure R is equally so on every figure contained in J? , and also that the sum, the difference, and, more generally, any linear combination of two additive functions of an interval that are of bounded variation on a figure, is itself of bounded variation on the same figure. An additive function whose values are of constant sign is termed monotone. A non-negative monotone additive function is also termed non-decreasing (for the same reason as in the case of non-negative additive functions of a set, cf. Chap. I, p. 8). Similarly, non-positive additive functions are also termed non-increasing. Every monotone additive function F of an interval on a figure R is clearly of bounded variation on R Q . If a function is of bounded variation on a figure # , its relative variations on R Q are evidently finite. Conversely, if for an additive function F of an interval on a figure J? , one or other of the two relative variations is finite, then both are finite, and therefore the 62 CHAPTER III. Functions of bounded variation. absolute variation is finite. For, if W(F; jR )<+oo say, then as W(jF;jR ) is the lower bound of the numbers F(R)=^F(R Q )F(R r 3R), where R is any figure contained in R Q , we find W(F; R ) ^ ^F(RQ) W(JP;/? )> oo. Moreover, this last inequality may also be written in the form F(R Q )^W(F' J R^)+W(F',R Q ). Here we replace F by F to derive the opposite inequality and then finally the equality F(R Q ) = W(Fj ^ )+W(J^; -R ). Hence any additive function of bounded variation is the sum of its two relative variations on any figure for which it is defined. This decomposition is termed the Jordan de- composition of an additive function of bounded variation, and is similar to the Jordan decomposition of an additive function of a set (Chap. I, 6). If F is an additive function of bounded variation of an inter- val on a figure fi , the three monotone functions defined for every figure R(2R Q by the relations ) and W 2 (JB)=W(^; JB), are likewise additive on R Q and are termed, respectively, absolute, upper, and lower variations of F. The first two are non-negative and the third non-positive. It therefore follows from the Jordan decomposition that every additive function of bounded variation on a figure is, on this figure, the difference of two non-decreasing functions. The converse is obvious. We shall now prove some elementary theorems concerning continuity properties of functions of bounded variation. (4.1) Theorem. If F is an additive function of an interval, of bounded variation on a figure K , (i) the series yoR tt (F;D n ) con- n verges for every sequence [D n ] of hyperplanes distinct from one another, and (ii) there is at most an enumerable infinity of hyperplanes D such that o Wll (/'; D)>0. Proof. In the proof of (i) we may clearly suppose that the hyperplanes D n are orthogonal to the same axis. Consider now the first k of these D n . We can associate with them, k non-overlapping intervals I\ y 1%, ..., //,, contained in R and such that o/? o (F; . Hence i n~ 1 /? 1 ; J? ) f I/ft, and, since ft is an arbitrary positive integer, ; D a ) l4] Functions of bounded variation. 53 To establish (ii), suppose that there is a non-enumerable in- finity of hyperplanes D such that o/ ?o (F;D)>0. There would then be a positive number f, such that o^ o (J T ;D)> for an infinity, (which would even be non-enumerable) of hyperplanes. But this clearly contradicts part (i) which has just been proved. We obtain at once from Theorem 4.1 the following (4.2) Theorem. For each additive function of an interval of bounded variation, there is at most an enumerable infinity of hyper- planes of discontinuity. (4.3) Theorem. If F is an additive function of cm interval of bounded variation on a figure R and we write W(I) W(F; J), the relations o/^jF 7 ; D)=0 and o Ho (Wj D)=Q are equivalent for every hyperplane D. Proof. Suppose, if possible, that (4.4) o* o (.F;D) = and (4.5) o* o (}f;D)>e for a hyperplane D and a number > 0. We shall show that it is then possible to define a sequence of figures {JK/,}/,^j,2,...> non-over- lapping, contained in R OJ and such that (4.6) #!>- and (4.7) \F(R n )\>e/2. To see this, suppose defined k non-overlapping figures R\^R^....Rk contained in jR , and let (4.6) and (4.7) hold for n=l, 2, ..., Tc. On account of (4.5) there is then an interval K^R such that I-R n =Q for n=l, 2, ..., fc, and such that W(jF; I) = W(I) >f. Hence, there exists a figure RCI such that |jF(lZ)|>W(-F; I)/2>*/2. Moreover since (4.4) asserts that F is continuous on R at D, we may sup- pose that R-D=0. But if we now choose R k +i = R, we see that the figure Rk+\ does not overlap any of the figures R n for l^n^i, and that (4.6) and (4.7) continue to hold for n=k+l. Having obtained our sequence {R n }j we conclude from (4.7) that W(F;R ):^2\F(R n )\=oo, and this contradicts our hypotheses. n The conditions (4.4) and (4.5) are thus incompatible, i. e. (4.4) implies o Ro (Wy D) Q. And since the converse is obvious, this completes the proof. From Theorems 4.3 and 3.1, we obtain at once the following (4.8) Theorem. In order that an additive function of bounded variation on a figure R be continuous on J? , it is necessary and suf- ficient that its three variations be so. 64 CHAPTER III. Functions of bounded variation. $ 5. Lebesgue-StieltJes integral. Lebesgue integral and measure. We need hardly point out the analogy between additive functions of bounded variation of an interval and additive functions of sets. This analogy will be made clearer and deeper in the present , by associating a function U* of a set with each addi- tive function U of bounded variation of an interval. In order to simplify the wording, we shall suppose that the functions of an interval are defined in the whole space. Suppose given in the first place, a non-negative additive function U of an interval; we then denote for any set E, by U*(E) the lower bound of the sums 2U(I k ), where {I k } is an arbitrary sequence of intervals such that E I*. For an arbitrary additive function U of bounded variation, with the upper and lower varia- tions W l and WM we denote by W* and ( W 2 )* the functions of a set that correspond to the non-negative functions W l and W 2 , and we write, by definition, U*=W* ( W 2 )*. The function U* is thus defined for all sets and is finite for bounded sets. When U is non-negative, U* is an outer measure in the sense of Carath^odory, i. e. fulfills the three conditions (Cj,), (C 2 ) and (C 3 ) of Chap. II, 4. Condition (C 3 ) is the only one requiring proof, the other two are obvious. Let therefore A and B be any two sets whose distance does not vanish, and let e be a positive number. There is then a sequence {I n } of intervals such that A+B(^^]In an d n e. We may clearly suppose that the intervals of the sequence have diameters less than Q(A,B), i. e. that none of them contains both points of A and points of B. We then have U*(A)+U*(B)^2U(InXU*(A+B)+e. This gives the in- n equality U*(A)+U*(B)*^U*(A+B) and establishes condition (C 3 ). The function [7*, determined by a non-negative function of an interval, itself determines, since it is an outer Carath6odory measure (cf. Chap. II, 4, p. 46), the class 2^* of the sets measurable with respect to U* and the process of integration (17*). To simplify the notation, we shall omit the asterisk and write simply Zu for Ct/*, integral (U) for integral (J7*), measure (U) of a set in- stead of measure (17*), fjdU instead of ffdU*, and so on. l5] Lebesgue-Stieltjes integral. 65 This slight change of notation cannot cause any confusion, since the measure 17* is uniquely determined by the function of an in- terval U. When U is a general additive function of an interval, of bounded variation, we shall understand by il# the common part of the classes % Wl and fi_ w , where W l and W 2 denote respectively the upper and lower variations of V. A function of a point /(#) will be termed integrable (U) on a set E, if /(#) is integrable (WJ and ( W 2 ) si- multaneously; by its integral (U) over E we shall mean the number ffdW l fjd( W a ), and we write it ffdU as in the case of K K E a non-negative function U. This integration with respect to an additive function of bounded variation of an interval is called Lebesgue-Stieltjes integration. In the case of the integra- b tion over an interval /=[, b] in J? 1? we frequently write f fdU a for ffdU. When the function U is continuous, every indefinite integral ( U) vanishes, together with the function ?/*, on the boundary of any figure. Consequently, an indefinite integral with respect to a continuous function U of bounded variation of an interval is additive not only as function of a set (</) but also as function of an interval. The most important case is that in which the given function of an interval H is the special function L (cf. 2, p. 59) that denotes the volume, of an interval. The outer measure (L) is also termed outer Lebesgue measure, and the integral (L), Lebesgue integral, while functions integrable (L) are often called, as originally, by Lebesgue, summable. The class of sets iii. will be denoted simply by 2. The outer measure (L) of an arbitrary set E is written meas<J57, and meas E without the suffix when E is measurable (8). We shall also denote this outer measure by \E\ or by *L(E) (or sometimes by L m (E) in K m ), thus extending to arbitrary sets the notation adopted for all figures It, since for the latter, as we shall see (cf. Th. 6.2), the measure (L) coincides with the values It(R)\R\. Finally, owing to the special part played by Lebesgue measure in the theory of integration and derivation, the terms "measure of a set", "measurable set", "measurable function", and 66 CHAPTER III. Functions of bounded variation. 60 on, will, in the sequel, be understood in the Lebesgue sense, whenever another sense has not been explicitly assigned to them. We also modify slightly the integral notation for a Lebesgue integral; and instead of ff(x)dL(x) we write ff(x)dx, or else E E Jf( x iy ii > x m) dx\ dxi ... dx m , when we wish to indicate the E number of dimensions of the space JK m under consideration. This brings us back to the classical notation. A special part, similar to that of Lebesgue measure in the theory of the integral, is played by Borel sets in the theory of ad- ditive classes of sets. In the first place, it follows from Theorem 7.4, Chap. II, that every class 2^, where U is an additive function of bounded variation of an interval, contains all the sets (33). In the sequel, we shall agree that additive functions of a set will always mean functions additive (53), unless there is explicit reference to another additive class of sets. Similarly, additive functions of a set that are absolutely continuous (53, L) or singular (53, L), will simply be called absolutely continuous or singular. In point of fact, Theorem 6.6 below, which asserts that every set measurable (fi) is the sum of a set (53) and a set of zero measure (L), will show that every additive function of a set, absolutely continuous (53, L), can be extended in a unique manner to all sets (2) so as to remain ab- solutely continuous (C, L). The special r61e of the measure (L) and of the sets C33) showed itself already during the growth of the theory. Lebesgue measure was the starting point for further extensions of the notions of measure and integral, whereas the Borel sets were the origin of general theories of additive classes and functions. The sets (93) were introduced, with measure (L) defined for them, by E. Borel [I, p. 46 50] in 1898. But it was not until some years later that H. Lebesgue [1; I], by simplifying and extending the definition of measure (L) to all sets (C), made clear the importance of this measure for the theory of integration and especially for that of derivation of functions. Vide E. Borel [1] and H. Lebesgue [6]. We have already seen in 1 of this book, how, by an apparently very slight modification of the classical definition of Riemann, we obtain the Lebesgue in- tegral. A similar remark may be made with regard to the relationship of Le- besgue measure to the earlier measure of Peano- Jordan. The outer measure of Peano-Jordan for a bounded set E is the lower bound of the numbers \I n \ n where {I n } is any finite system of intervals covering E. Lebesgue' s happy idea was to replace in this definition, the finite systems of intervals by enumer- able ones. [6] Measure defined by a function of an interval. 7 We have given in the text a more general form to Lebesgue's definition, relative to an arbitrary non -negative function of an interval. This relativizing of Lebesgue measure is due to J. Radon [1] and to Ch. J. de la Vallee -Poussin [1; I]. The parallel extension of the Lebesgue integral is also due to J. Radon. In the text we have termed it Lebesgue -Stieltjes integral; it is sometimes also termed Lebesgue- Radon integral or Radon integral. For a systematic exposition of the properties of this integral, vide H. Lebesgue [II, Chap. XI]. A particularly interesting generalization of the Lebesgue integral, of the Stieltjes type, has been given by N. Bary and D. Menchoff [1]; it differs considerably from the other generalizations of this type. Finally, for an account of the Riemann -Stieltjes integral (which we shall not discuss in this volume) vide W. H. Young [2], S. Pollard [1], R. C. Young [1], M. Fr^chet [51 and G. Fichtenholz [2]. It was again J. Radon [I, p. 1] who pointed out the importance of the Lebesgue -Stieltjes integral for certain classical parts of Analysis, particularly for potential theory. The modern progress of this theory, which is bound up with the theory of subharraonic functions, has shown up still further the fruitfulness of the Lebesgue -Stieltjes integral in this branch of Analysis (cf. the memoirs of F. Riesz [4] and G. C. Evans [1]). 6. Measure defined by a non-negative additive function of an interval. In this , U will denote a fixed non- negative additive function of an interval. In the preceding we made correspond to any such a function [7, an outer Carath^o- dory measure U*. Besides the properties established in Chap. II for all Carath^odory measures, the function of a set U* possesses a number of elementary properties of a more special kind which we shall investigate in this . (6.1) Lemma. If D denotes a hyperplane or a degenerate interval, the relation o(U;D) = Q implies U*(D) = Q. Proof. Since every hyperplane is the sum of a sequence of degenerate intervals (cf. 2, p. 57), it is enough to prove the lemma in the case in which D is a degenerate interval [ai, fti; a;, ft*; ...; a m , b m ]. Let R be a cube containing D in its interior, an arbitrary open set such that DC#> and tet D f = [a l J b l +f J a 2 f, b 2 +t; ...; a m f, b m +e], where e is any positive number, sufficiently small to ensure that D e (2R-G. Since D e is then an ordinary closed interval con- taining D in its interior, we find ff*(D)<Z7(D ) <O(tf; R.Q), whence U*(D)^o R (U; D) = 0. 68 CHAPTER III. Functions of bounded variation. (6.2) Theorem. For every figure R we have (6.3) U*(R) < U ( R)^ U*(R), and, if the oscillation of U at the boundary of R vanishes, (6.4) U*(R)=U(R) = In particular therefore, if U is a continuous function, the equal- ity (6.4) holds for every figure R. Proof. In virtue of Theorems 2.2 and 4.2 the set R is ex- pressible as the sum of a sequence of non-overlapping cubes {/*} such that the oscillation of U vanishes at all faces of all the I*. Hence, by the preceding lemma U*(R )=U*(I it)j and since *(Ik) for each positive integer n, we get *^l A-i )^ U(R). To establish that U(R)^U*(R), it is enough to show. that for every sequence of intervals {/*} such that h k Now, if {I k } is such a sequence, we have, by the well-known covering # theorem of Borel-Lebesgue, -KdJj ^* f r some sufficiently large *=i N N value of N. Hence V(RX2U(ROI*)^U(I*). k-i k- \ Finally, denoting by B the boundary of R, let us suppose that o(U;B) = 0. It then follows from Lemma 6.1 that J7*(JB)=0, so that J7*(jR) = Z7%R)> and the equality (6.4) follows at once from (6.3). (6.5) Theorem. Given an arbitrary set E and any positive e, there is (i) an open set G such that #C# <*>nd U*(G)^U*(E)+e, (ii) a set He<& 6 such that ErH and U*(H)=V*(E). Proof, re (i). There exists for each e>0, a sequence of inter- vals {/} such that EC2ft and that U(InXU*(E)+f. Hence, n n writing G=l n , we find, on account of condition (C 2 ) of Carath^odory n (Chap. II, p. 43) and Theorem 6.2, that Z [6] Measure defined by a function of an interval. $9 t'e (ii). Let us make correspond to Z?, for each positive integer w, an open set G n containing E and such that U*(G n )^U*(E) + I/n; this is always possible by (i). The set H=[]G n clearly fulfills our n requirements and this completes the proof. Every set ((5d) is of course measurable (fit/). Hence it follows at once from Theorem 6.5 that every set is regular (cf. Chap. II, 6) with respect to the outer measure U*, and therefore that this measure is itself regular. (6.6) Theorem. Each of the following conditions is necessary and sufficient for a set E to be measurable (fit/): (i) for every >0 there is an open set G^)E such that (ii) there is a set (C&a) containing E and differing from E at most by a set of measure (U) zero\ (iii) for every e>0 there is a closed set F(^E such that (iv) there is a set ($) contained in E and differing from E at most by a set of measure ( U) zero. Proof. We shall first prove all these conditions necessary. Let E be a set measurable (fit/) and e a positive number. We begin by representing E as the sum of a sequence {E n } n _\^. of sets measur- able (8c/) of finite measure; we may write for instance, B ll =JS-8(0;n). This being so, we associate with each set E n , in accordance with The- orem 6.5, an open set G n ^E n such that U*(G n )^U*(E n )+fl'2 n . Hence, the sets E n being measurable (fit/), we have U*(G n E n )^(l2" for every n, and if we write G=G n9 we find EC and U*(G and this proves condition (i) necessary. To prove the necessity of condition (ii), we attach to the given set E measurable (fit/) a sequence {Q n } of open sets such that, ECQn and U*(Q H E)^l/n for each n. Writing #=//#, we see that //t d , ECU and U*(HE) = 0. Finally, we observe that for any set 4, the relation A^)CE implies UACE and E CA=A CE; further, if A is a set () or (,>), the set CA is a set (5) or ($) respectively. Hence every set E measurable (fit/) fulfills conditions (iii) and (iv), since, by the results just proved, its complement ?E fulfills conditions (i) and (ii). 70 CHAPTER III. Functions of bounded variation. The sufficiency of conditions (ii) and (iv) is evident, since sets of measure ( U) zero, and sets (<>) or ($<,), are always measur- able (fii,). To establish the suffic ; ency of conditions (i) and (iii), we need only observe that they imply respectively conditions (ii) and (iv). Thus, for instance, if (iii) holds, there is for each positive integer n a closed set F n (^E such that U*(E F n )^lln. The set is therefore a set (a) contained in E and such that From Theorem 6.6 it follows in particular that the general form of a set measurable (#) is B+N, where B is a set measurable (93) and N a set of measure (17) zero. In other words, %u is the smallest additive class containing the Borel sets and the sets of measure (V) zero. It follows that 2OC whenever the function of an interval 17 is absolutely continuous (vide 12). (6.7) Theorem. For any set E there is a set H<$>A containing E and such that (6.8) U*(H.X)=V*(E-X) for every set X measurable (fie,). Proof. It is enough to show that there is a set H^)E measurable (i*f/) for which (6.8) holds. For, by Theorem 6.6 we can always en- close such a set H in a set (<j) differing from it by a set of measure (V) zero. Let us represent E as limit of an ascending sequence (E n \ of bounded sets, which are therefore of finite outer measure (C7); and let us associate, as we may by Theorem 6.5, with each E n a set H n <$> 6 such that E n CH n and U*(H n )=U*(E n ). Then, for every set X measurable (fit/), ffVH n .*)=Z7*(ff n )^l7*(l^^ from which, writing ff=liminf ff n , we deduce by means of The- n orem 9.1 of Chap. I, that U*(H-X) ^ lim inf U*(H n .X) ^ lim U*(E n -X) ^ V*(E.X), n n and this implies (6.8), since H lim inf H n 3) lim E n E. n n In the theorems of this we have supposed the function U of an interval to be non-negative. But, by slight changes in the wording, l6] Measure defined by a function of an interval. 71 the theorems can easily be extended to arbitrary functions of bounded variation. As an example we mention the following theorem which corresponds to Theorem 6.5: (6.9) Theorem. If F is an additive function of bounded variation of an interval, then for any bounded set E and any e>0 there is an open set G^)E such that \F*(X) F*(E) ^e for every bounded set X satisfying the condition Proof. Denoting by W l and W 2 two functions of an interval that are respectively the upper and the lower variation of F, we can, by Theorem 6.5, enclose E in each of two open sets G and # 2 such that W7(<?i)^TPf(jE;)+ and W$(Q 2 )^W$(E) f. Therefore, writing G=G l -G 2 we have E(^G, and for any bounded set X such that ECXCQj w find O^Wf(X) Wf(E)^e and ) W$(X)^c, whence by subtraction \F*(X)F*(E)\<^f. Let us still prove a theorem which allows us to regard all non- negative additive functions of a set in lt m as determined by non- negative additive functions of an interval. We recall that, according to the conventions of 3, p. 59 and 5, p. 66, we always mean by additive functions of a set, functions of a set that are additive (93) on every figure. (6.10) Theorem. Given any non-negative additive function of a set, there is always a non-negative additive function F of an interval such that <D(X)=F*(X) for every bounded set X measurable (93). Proof. Let us denote for each interval 7 [ai, b\, ...; a mj 6 m ], by 7 the interval (ai, fti; ...; a mj b m ] half open to the left, and let us define the non-negative additive function of an interval by writing F(l) = f I>(I} for every interval 7. This being so, we observe that any bounded open set G can be expressed (cf. Theorems 2.2 and 4.2) as the sum of a sequence {!} of non-overlapping intervals at whose faces the oscillation of F vanishes; and therefore by Theorem 6. 2, $(G)=2<D(T n )=2F(I n ) = I*)=F*(Q). Thus the equation Q(X)=F*(X) holds whenever^ /i is a bounded open set, and therefore also whenever X is a bounded set (a), since the latter is expressible as the limit of a descending sequence of bounded open sets. It follows further that $(X) = = F*(Jf)=0 for every bounded set X of measure (F) zero, since, 72 CHAPTER III. Functions of bounded variation. by Theorem 6.6, such a set X can be enclosed in a bounded set ((&<*) ot measure (F) zero. This completes the proof, since every bounded set (93) is, by Theorem 6.6, the difference of a bounded set (a) and of a set of measure (F) zero. The proof of Theorem 6.10 could also be attached to the following general theorem concerning functions additive (93) defined on any metrical space M\ if two such functions coincide for every open set, they are identical for all sets ("B). This theorem is easily proved. 7, Theorems of Lusin and Vftali-Carathodory. We shall establish in this two theorems concerning the approximation to measurable functions by continuous functions and by semi- continuous functions. As in the preceding , U will stand for a non- negative additive function of an interval, fixed in any manner for the space li m . (7.1) Lusiris Theorem. In order that a function /(#), finite on a set E, be measurable (2u) on E, it is necessary and sufficient that for every e >0, there exists a closed setFQE such that U*(E jF)<f, and on which f(x) is continuous. Proof. To show the condition necessary, we suppose f(x) finite and measurable (Zu) on E, and we deal first with two partic- ular cases: (i) f(x) is a simple function on E. The set E is, in this case, the sum of a finite sequence E\, E 2j ..., E n of sets measurable (2u) no two of which have common points, such that f(x) is con- stant on each of these sets. By Theorem 6.6 there exists, for each set E t j a closed set FtCEi such that U*(E t F t )</n. Writing /i F=2F h we then have F(^E and V*(E F)<e, and moreover /~t the function f(x) is clearly continuous on F. (ii) E is a set of finite measure (U). In this case, by Theorem 7.4, Chap. I, (applied separately to the non-negative and to the non-positive parts of /(#)), there is a sequence {/n(#)},i=i,2,... of simple functions, finite and measurable (2^), that converges on E to f(x). By Egoroffs Theorem (Chap. I, Th. 9.6), this sequence converges uniformly on a set P^E, measurable (2v) and such that U*(E P)<f/2. This set P may further be supposed to be closed, on account of Theorem 6.6. Finally, by (i) we can attach to each function /(#), a closed set P n CE such that U*(EP n )<t/2 n+l , [7] Theorems of Lusin and Vitali-Carath^odory. 73 and on which f n (x) is continuous. Hence, writing -F=P/"/Pn> we get U*(EF)^U*(EP) + SU*(EP n )^t; and moreover, all n the /(#), and therefore also the function f(x) = lim /,,(#), are con- n tinuous on F, a set which is evidently closed. We now come to the general case where E is any set measur- able (8tf). Let E n =E.(8n8 n -i), where S =0 and S n =S(Q;n) for n^l. By (ii), there exists, for each n^l, a closed set QnC^n such that U*(E n Q n )<e/2 n , and on which f(x) is continuous. Writing F=yQ n , the set F is closed, we have U*(EF)^ n^l ^U*(E n Q n )<fj and f(x) is continuous on I' 7 . n The proof of the necessity of the condition is thus complete. Let us now suppose, conversely, that the condition is satisfied. The set E is then expressible as the sum of a set N of measure ( U) zero and of a sequence {F n } of closed sets on each of which f(x) is continuous. The function / is thus measurable (fit/) on N and on each of the sets F n (cf. Chap. II, Th. 7.6), and therefore on the whole set E. For the various proofs of Lusin's theorem, vide N. Lusin [1J, W. Sier- pifiski [6] and L. W. Cohen [1]. (7.2) Lemma. Given a junction /(#), measurable (2u) and non- negative in the space JR mj there exists , for each e>0, a lower semi- continuous function h(x) such that (7.3) h(x)^f(x) at each point #, and (7.4) f[h(x)f(x)]dV(x)^e Rm (where, in accordance with the convention of Chap. I, p. 6, the difference h (x) f(x) is to be understood to vanish at any point x for which h(x}=f(x) = Proof, (i) First suppose that f(x) is bounded and vanishes outside a bounded set E measurable (C*/). Let ri=e/[l + U*(E)]. We write E k =E[xcE; (fc I)i?</(o?)<fci7] for *=1, 2, ... and we X associate with each set E k an open set (7*31?* such that 74 CHAPTER III. Functions of bounded variation. (7.5) V*(G k Further, denoting by c k (x) the characteristic function of GA, we oo write h(x)=kt]-Ck(x). Since each function c k (x) is evidently lower A=l semi-continuous, the function h (x) is so too. We also observe that h(x) fulfills condition (7.3). On the other hand R m ^i H m Jr=l whence, by (7.5), we obtain ff(x)dU(x)+e. m in From this, remembering that f(x) is integrable on R m , (7.4) follows at once. (ii) We now pass to the general case and represent firstly f(x) 00 in the form f(x)=2f n (x), where the f n (x) are bounded non-negative n=l functions measurable (C*/), each of which vanishes outside a bounded set. We may do this, for instance, by writing /(#) = *(#) *n-\(n) where s n (x) = f(x) for (*(Q,x)<n and n for (>(0, x) < n and f(v) > w, for n-0, 1,2, ... By what has t>een proved in (i), there exists for each function f n (x) a lower semi-continuous function h n (x) such that h n (x)^f n (x) at every point a?, and that f[h n (x) f n (x)]dU(x) <e/2". The function Urn oo h(x)=]?h n (x) is then evidently lower semi-continuous and fulfills con- Km and this completes the proof. [7] Theorems of Lusin and Vitali-Carath6odory. 75 (7.6) Theorem of Vitali-Carathodory. Given a junction f(x) measurable (2u) in the space JR m , there exist two monotone sequences of functions {?(#)) and (u n (x)} for which the following conditions are satisfied: (i) the functions l n are lower semi-continuous and the functions u n are upper semi-continuous, (ii) each of the functions l n is bounded below and each of the functions u n is bounded above, (iii) the sequence {l n } is non-increasing and the sequence {u n } is non-decreasing, (iv) l n (x)^f(x)^u n (x) for every x, (v) liml n (x)= f(x) = \im u n (x)' almost everywhere (U), n n (vi) on every set E on which f(x) is integrable (U), so are the functions l n (oo) and u n (x) and we have lim f l n (x) dU(x)=\im fu n (x) dU(x)= f'f(x) dU(x). " E n E E Proof. By expressing the function f(x) as the sum of its o non-negative and non-positive parts f(x) and f(x) (Chap. I, 7), o we may suppose that f(x) is of constant sign, say non-negative. By the preceding lemma, we can associate with f(x) a sequence of lower semi-continuous functions {h n (x)} n =-.\.*,.. such that h n (x)^f(x) for every x and (7.7) Umf[h n (x)f(x)]dU(x) = 0. Writing l n (x) = min [hi(x), h%(x), ..., h n (x)] we therefore obtain a non- increasing sequence of lower semi-continuous functions (l n (x)} that evidently fulfills conditions (i), (ii), (iii) and (iv); moreover, it follows from (7.7) that lim / [l n (x) f(x)]dU(x) 0, and hence n that the functions l n (x) fulfill also conditions (v) and (vi). In order now to define the sequence (u n (x)}, we attach to the function I// (x) a non-increasing sequence of lower semi-continuous functions {</(#)} such that Mmg n (x) = l/f(x) almost everywhere ( U). n Such a sequence certainly exists by what has just been proved. The functions l/g n (%) then form a non-decreasing sequence of upper semi-continuous functions, that converges almost everywhere (U) to f(x). If we now write u n (x) llg n (x) when l/g n (x)^n, and 76 CHAPTER III. Functions of bounded variation. u n (x) n when !/</(#)> n, we obtain a sequence (u n (x)} of bounded functions with the same properties, which therefore satisfies con- ditions (i v). Finally, since the functions u n (x) are non-negative, we can apply Lebesgue's Theorem (Chap. I, Th. 12.6) to derive from (iii) and (v) that lim fu n (x)dU(x) = ff(x)dU(x) on every " E E set E measurable (2t;), and this implies (vi). Conditions (i) and (v) of Theorem 7.6 imply that every function measur- able (u) is almost everywhere (V) the limit of a convergent sequence (with finite or infinite limit) of semi-continuous functions, and thus coincides almost everywhere (U) with a function of the second class of Baire. This result, due to G. Vitali [2] (cf. also W. Sierpinski [6]) was completed by C. Carath^odory [I, p. 406], who established for every measurable function f(x) the existence of two sequences of functions fulfilling conditions (i) (v). Condition (vi), which includes, as we shall see later, the theorem of de la Vallee Poussin and Perron on the existence, for summable functions, of ma jo rant and minorant functions, has been added here because its proof is naturally related to those of conditions (i v). There is an obvious analogy between the property of measurable functions expressed by the theorem of Vitali-Caratheodory, and the properties of meas- urable sets stated in conditions (i) and (iii) of Theorem 6.6. By taking into ac- count the geometrical definition of the integral (cf. below 10), we might even base the proof of Theorem 7.6 directly on Theorem 6.6 (vide the first ed. of this book, pp. 88 91). $ 8. Theorem of Fubini. Given two Euclidean spaces R p and R q , if x(a h a 2 , , p) and y=(0 p +i, Vt-2, > a p+g) are two points situated respectively in these two spaces, we shall denote by (Xj y) the point (aj, a 2 , ..., a p + q ) in the space R p -\- q . ^ X and Y are two sets situated respectively in the spaces R p and R qj we shall denote by Xx Y the set of all points (#, y) in R p + q such that xeX and t/f Y. In particular, if X and Y are two intervals closed, open, or half open on the same side X x Y also is an interval in R p + q , which is closed, open, or half open on the same side as X and Y. Every interval I=[ai, 61; ...; <*>/>+< Vf</l can evidently be expressed and in a unique manner in the form IiXl 2 where 1^ and I 2 are intervals in lt p and R q respectively; we merely have to write I\ = [a\j b\; ...; a p9 b p ] and / 2 =[a p +i, VM; ...; fi p + q , bp+ q ]. Given two additive functions of an interval, V and F, in the spaces R p and R q respectively, we determine a function of an inter- val T in R p + q by writing T(Z 1 xI 2 )=f/(/ 1 )-F(/ 2 ) for each pair of intervals I\C,R P and I 2 C^- The function T thus defined, clearly additive when U and V are, will be denoted by UV. In particular, q [8J Theorem of Fubini. 77 we see easily that Lp+^LpL^, where L p , L 7 and L p+<? denote the volume in the spaces R PJ R q and JRf>+ q respectively (cf . 2, p. 59). It is known since Cauchy that, if I I and I 2 are respectively two intervals in the spaces R p and R q , integration of any continuous function over the interval I\xl2dR P + Q may be reduced to two successive integrations over the intervals I I and 7 2 . By repeating the process, any integral of a continuous function on an ra-dimensional interval may be reduced to m successive integrations on linear intervals in R v . This classical theorem was extended by H. Lebes- gue [1] to functions measurable (fi) that are bounded, and then by G. Fubini [1] (cf. also L. Tonelli[2]) to all functions integrable (L), whether bounded or not. We shall state this result in the follow- ing form: (8.1) Fubini's Theorem. Suppose given two non-negative ad- ditive functions U and V of an interval in the spaces R p and R q respectively , and let f(x,y) be a non-negative function measurable (2uv) in R p + q . Then (h) f( x ^y) i 8 a function of x, measurable (%u) in R p for every yeR qj except at most a set of measure (V) zero, (i%) f( x j y) is a function of y, measurable (2v) in R q for every xeRp, except at most a set of measure (U) zero, (ii) ff(x, y) dVV(x, y) - / [ff(x, y) dU(x)] dV(y) = Kp+ti K( t R P = f[ff(x,y)dV(y)]dU(x). *P", Proof. Let us write for short, T=UV. By symmetry, it is enough to show that every non-negative function /(#, y) measurable (fir) in Rp+q fulfills condition (i^ and also the relation (8.2) ff(x, y) dT(x, y)=/ [ff(x, y) dV(x)] dV(y). **p+q Rq Kp For brevity, we shall say that a function f(x,y) in R P + q has the property (jP), if it is non-negative and measurable (fir) in R P + q > and if it fulfills condition (i x ) and the relation (8.2). For the sake of clearness, the reasoning that follows is divided into a several auxiliary propositions. 78 CHAPTER III. Functions of bounded variation. (8.3) The sum of two functions with the property (F), and the limit of any non-decreasing sequence of such functions, have the prop- erty (F). Also, the difference of two functions with the property (F), has the property (F), provided that it is non-negative and that one at least of the given functions is finite and integrable (T) on the space For the sum, and for the difference, of two functions, the statement is obvious. Let therefore \h n (x, y)} be a non-decreasing sequence of functions in tl p ^ q having the property (F), and let h(x, y) == lim h n (x, y). The definite integrals j h n (x, y)dU(x) exist, *, and constitute a non-decreasing sequence, for every y f ll q except at most those of a set of measure (V) zero. Consequently, by Lebesgue's theorem on integration of monotone sequences of functions: fh(x,y) dT(x,y)=]imfh n (x, y) dT(x, y)=lim/ [/*(*, y) dU(x)] dV(y)^ p+q "Rp+l "*** **P = f [lim fh n (x, y) dU(x) ] dV(y) - /' [/ h(x, y) dU(x) ) dV(y), and this establishes the property (F) for the function h(Xjy). (8.4) The characteristic function of any set E(21i p + q measurable has the property (F). We shall establish this, first for very special sets E and then, by successive stages, for general measurable sets. Suppose in the first place that 1 E=AxB, where A and B are intervals half open to the left, situated respectively in R p and liq, and such that the oscillations of U and of V vanish at the bound- aries of A and B respectively (cf. 3, p. 60). The oscillation of the function TUV therefore vanishes at the boundary of the interval E=AxB, and we find by Theorem 6.2 (8.5) On the other hand, for every y e R q the function CE(X, y) is in x the characteristic function either of the half open inter- val J., or of the empty set, according as yeB or y R q B. This function is therefore measurable (2^), and indeed measurable (53), for every yeltg, and, by (8.5) [8] Theorem of Fubini. 79 fc s (x, y) dT(x, y)=T*(E)= U*(A) . V*(B)=f [fc s (x, y) dU(x)] dV(y). R q K P 2 E is an open set. We shall begin by showing that, in this case, E is the sum of a sequence of half open intervals {!} no two of which have common points, these intervals I n being of the form A n xB n where (a) A n and B n are intervals, half open to the left, situated in H p and R q respectively, and (b) the oscillations of U and V vanish at the boundaries of A n and B n respectively. To see this, let (U (A) | be a regular sequence of nets in Jf p formed of intervals half open to the left and such that the oscillation of U vanishes at the boundary of each of these intervals; by Theorems 4.2 and 2.1, such a sequence certainly exists. And let {93 (A) } be a sequence of nets similarly constructed for the space li q and for the function V. We denote, for each ft, by Z (k) the system of all half open intervals in R p + q which are of the form A xB where AeU (k) and B^ (k) . The systems of intervals Z (ft \ thus defined, form a reg- ular sequence of nets of half open intervals in the space It p + q . The set E being open, we can therefore express it (cf. 2, p. 58) as the sum of a sequence of half open intervals {!} taken from the nets 3^ and without points in common to any two. We see at once that each interval / of this sequence is of the form A n xB n where A n and B n satisfy conditions (a) and (b). This being so, we have CE(X, y) J^c/ n (x, y), where on account 71 of the result established for the case 1, each of the characteristic functions c/ /f (o?, y) has the property (F). Therefore, to verify that the function c#(#, y) also has this property, we need only apply (8.3). 3 E is a set (a). First suppose that, besides, the set E is bounded. E is then the limit of a descending sequence of bounded open sets ((?}. The functions C0 t (#y) C0 n (#, y) constitute a non- decreasing sequence of non-negative functions which have, by 2 and (8.3), the property (F). Consequently, again on account of (8.3), the limit function of this sequence h(x, y) = C3 l (x,y) MO?, y) itself has the property (F) and the same is therefore true of the function c*(0, y) = c0,(0, y)h(x, y). Now if E is an arbitrary set (CM, we can express it as the limit of an ascending sequence {H n } of bounded sets (a). By what has 80 CHAPTER III. Functions of bounded variation. just been proved, the characteristic functions of the sets H n have the property (F) and, consequently, the function c#(#,y)=limc// n (#, y) itself has- the property (F). " 4 E is a set of measure (T) zero. There is then, bj Theorem 6.5, a set He($>d containing E and of measure (T) zero. By the result established for sets (C&a), the function G H (x 1 y) has the property (F), and therefore J [j c// (x, y) dU(x)^dV(y) = K q p =JcH(x,y)dT(x,y)=T*(H) = 0. Hence, for every yeR g , except at Rp+q most a set Y of measure (F) zero, c H (x,y)dU(x)=Q, i.e. as function of a?, vanishes almost everywhere ( U) in R p . Hence, a fortiori, C(#,y)^Ic//(#,t/) as function of #, vanishes almost every- where ((7), and is consequently measurable (fit/), for all yeR q , except at most for those of the set Y of measure ( V) zero. Finally, we clearly have f[fc E (x J y)dU(x)]dV(y)=Q = T*(E)=fc E (x,y)dT(x,y). ~ The function c,(#, y) thus has the property (F). On account of Theorem 6.6 every set E measurable (Sir) is expressible in the form E=H Q, where H is a set (,5) and Q is a set of measure (T) zero contained in H. We thus have c^(#, y) = = e H (x,y) CQ(#, y), and by (8.3) the proposition (8.4) reduces to the special cases 3 and 4 already treated. The proposition (8.4) being thus established, let /(#, y) be any non-negative function measurable (2r) in the space R p + q . By Theorem 7.4, Chap. T, the function / is the limit of a non-decreasing sequence of simple functions, finite, non-negative, and measurable (2r). Now each of these simple functions is a linear combination, with positive coefficients, of a finite number of characteristic functions of sets measurable (2r), and therefore has the property (F) on ac- count of (8.4). Thus the function / is the limit of a non- decreasing sequence of functions with the property (F), and so, by (8.3), / itself has the property (F). This completes the proof of Theorem 8.1. Let us make special mention of the particular case of the theorem in which f(x,y) is the characteristic function of a meas- urable set: [f 8] Theorem of Fubini. gl (8.6) Theorem. If U and V are two non-negative additive functions of an interval in the spaces R p and R q respectively, and if Q is a set measurable (2uv) in the space Ji> p + qj then (\i) the set E[(#, y)eQ] is measurable (%u) for every yeR q , X except at most a set of measure (V) zero, (i 2 ) the set E[(#, y)eQ] is measurable (Sv) for every xeR PJ y except at most a set of measure (U) zero, and (ii) the measure (UV) of Q is equal to f J7* {E [(a?, y) e Q]} dV(y) = f F* {B [(a?, y) e Q]} dU(x). *, * *, " Pubini's theorem is frequently stated in the following form: (8.7) Theorem. Let U and V be two additive functions of bounded variation of an interval in the spaces R p and R q respectively. Then for every function f(x,y) integrable (VV) on Rp+q, the relation (ii) of theorem 8.1 holds good and the function f(x,y) is integrable (U) in x on JK P for every yeJK qj except at most a set of measure (V) zero, and integrable ( V) in y on R q for every x e lt p , except at most a set of measure (U) zero. We reduce this statement at once to that of Theorem 8.1 by expressing the function / as the sum of its non-negative and non- positive parts, and by applying to the functions of an interval U and V the Jordan decomposition ( 4, p. 62). Further generalizations of Fubini's theorem for the Lebesgue -Stieltjes integration (in particular including the theorems analogous to Theorem 15.1 of Chap. I) were studied by L. C. Young in his Fellowship Dissertation (Cam- bridge 1931, unpublished). An account of these researches will be given in the bool The theory of Stieltjes integrals and distribution-functions by L.C. Young (Oxford, Clarendon Press). It, follows in particular from Theorem 8.6 that for any set Q measurable in the sense of Lebesgue in the space Rp+q, its measure (Lp+q) is given by the definite integrals fl* P {E [(x.y)cQ]}dI* g (y)=* f l. Q {E[(x J y)eQ]}dLp(x). It is never- *< ' , ' theless to be remarked that the existence of these two integrals does not in general enable us to draw any conclusion as to measurability (C) of the set Q. W. Sierpinski [6] has in fact constructed in the plane a set non -measurable (i>) having exactly one point in common with every parallel to the axes. This con- struction depends, needless to say, on the axiom of selection of Zermelo. For an interesting discussion of Fubini's theorem for Lebesgue integration of functions of variable sign, vide G. Fichtenholz [1], 82 CHAPTER III. Functions of bounded variation. We complete the theorems of this by the following (8.8) Theorem. If Q is a set measurable (93) in the space the set E[(#, y) cQ] is mewurable (93) in the space R p for every y e R qj X and the set E[(#, y)eQ] is measurable (93) in R q for every xeR p . y Similarly, if a function /(#,y) is measurable (93) in the space JR p + q , then in R p the junction f(x,y) is measurable (93) in x for every yelt q , and in R q the junction /(a?, y) is measurable (93) in y for every xeJK p . Proof. It will be enough to prove the first half of the theorem, since the second half obviously follows from the first. Let us denote by 93 the class of all sets Q in R p+g such that the sets E[(o?, j/)t#], for every y*R q , and the sets E[(#, y)e$], for every xeRp, are y measurable (93) in the spaces R p and R q respectively. If a set QdR P + q is closed, so are the sets E[(#, y)*Q} and E[(#, y)eQ]. The class 93 x H thus contains all closed sets of the space R p + q , and on the other hand we see at once that 93 is additive. It follows that 93 includes all Borel sets in the space R p + q (cf. the definition, Chap. II, p. 41), and this completes the proof. * 9. Fubini's theorem In abstract spaces. We shall return in this to the abstract considerations of Chap. I and show that for abstract spaces, theorems similar to those of the preceding hold good. Given any two sets X and Y, we shall denote by X x Y the set of all pairs of elements (x, y) for which x<-X and yeY. The set X X Y is often called combinatory product or Cartesian product (cf. C. Kuratowski [I, p. 7]) of the sets X and Y. The following identities are obvious (9.1) (X l xY l )-(X 2 (9.2) (jr f x r,) (*ix r the sets X 19 Z 2 , Y 19 Y 2 being quite arbitrary. If 9C and ?) are additive classes of sets in the spaces -Y and V respectively, 9?) will denote the smallest additive class of sets in the space Xx V, containing all product-sets of the form Xx Y, where Ze9t and Ye?). [9] Fubini's theorem in abstract spaces. 83 For auxiliary purposes, we shall make use in this of the following definition: a class 91 of sets witi be termed normal, if (i) the sum of every sequence of sets (91) no two of which have common points is itself a set (9t) and (ii) the limit of every descending sequence of sets (91) is a set (91). We shall begin by proving the following analogue of Theorem 8.8: (9.3) Theorem. Let 9 and 9) be two additive classes of sets in the spaces X and Y respectively. Then, if Q is a set measurable (9E?)) in the space Xx Y, the set E[(#, y)*-Q] is measurable (9E) for every X */el r , and the set 'K[(x 9 y)eQ] is measurable (?)) for every xtX. i/ In the same way a function /(#, y) which is measurable (SE?)) in the space Xx Y, is measurable (SE) in x for every y cY and meas- urable ( < 3)) in y for every XfX. Proof. It is enough to prove only the first part concerning sets. To do this, we denote by SPI the class of all sets Q in Xx Y such that the set E[(.r, y)eQ] is measurable (9C) for every yeY, X and that the set B[(#,y)e()] is measurable ( < Z)) for every xeX We see y at once that the class 9Jt is additive in the space Xx Y and that, besides, it includes all sets Xx Y for which XeSE and Y %). Hence SE ?) C 2ft > and this completes the proof. Before proceeding further we shall establish the following lemma: (9.4) Lemma. If 2E and ?) are two additive classes of sets in the spaces X and Y respectively, the class 9E?) coincides with the smal- lest normal class that includes the sets X x Y for which Xe SE and Y 6 ( 2). Proof. -For brevity let us term elementary any set JfxY for which X <- SE and <?), and let 91 denote the smallest normal class which includes all elementary sets (i. e. the common part of all the normal classes that include these sets). Clearly 91 C$?) since SE?) is also a normal class. In order to establish the opposite inclusion, it is enough to prove that the class 9J is additive, and this will be an immediate consequence of the following two properties of the class 9i : 84 CHAPTER III. Functions of bounded variation. (9.5) The common part of any sequence of sets (9t ) is itself a set (9t ). (9.6) The complement (with respect to the space Xx Y) of any set (9t ) is again a set (9t ). To prove (9.5), it is enough, since the class 9t is normal, to show that the common part of two sets (9t ) is a set (9t ). For this purpose, let 9^ be the class of all the sets (9t ) whose common parts with every elementary set belong to 9t . From the identity (9.1), it follows that the common part of two elementary sets is an elementary set, and hence that 9^ includes all the elementary sets. On the other hand, we verify at once that 5Rj is a normal class. This gives 97 C9ti> and since by definition 9tiC9t , we obtain 91 1 =97 . Let now 9? 2 be the class of all the sets (9? ) whose common parts with every set (9t ) belong to 9t . Since 9? i: =:9t , the class 9t 2 includes all elementary sets. Furthermore, 9I 2 is clearly a normal class. We therefore have 9t 2 ^ o> an( i this proves (9.5). To establish (9.6), let 9t 3 be the class of all the sets (9t ) whose complements are also sets (9t ). On account of the identity (9.2) the complement of any elementary set is the sum of two elementary sets without common points, and so, a set (9J ). Therefore the class 9? 3 includes all elementary sets and, to conclude that 91 3 97 , it suffices to show that the class 9i 3 is normal. Let therefore {X n } be any sequence of sets (9t 3 ) without com- mon points to any two of them, and let X be the sum of the se- quence. The set X clearly belongs to the class 9? . On the other hand, the sets CX n are, by hypothesis, sets (9? ); so that, by (9.5), the same is true of their product GX\[CX n . Thus we have at n the same time, Xeyi Q and (LTe^o, and therefore Xf97 3 . Again, let {Y n } n ^i,2,... be a descending sequence of sets (9? 3 ), and Y its limit. The set Y clearly belongs to the class 97 . On the other hand, consider the identity and observe that no two of the sets CYi and Y n >CY n +i for w=l,2, ... have common points. Since these sets belong, by (9.5), to the class 9t , so does the set CY. Thus we have both re9! and CYe9} > whence Ye97 3 . The class 9I 3 is therefore normal, and this establishes (9.6) and completes the proof of Lemma 9.4. [9] Fubini's theorem in abstract spaces. 35 We can restate Lemma 9.4 in the following more general form: (9.7) Given in an abstract space T a class Q of sets additive in the weak sense, then the smallest class that is additive (in the complete sense) and contains Q. coincides with the smallest normal class containing Q. The proof is the same as for Lemma 9.4. If $ and ^ are additive classes in the spaces -Y and I' respectively, the finite sums of the sets XxY for which Jfe9E and Ye9), constitute, according to formulae (9.1) and (9.2), a class that is additive in the weak sense (vide Chap. I, p. 7) in the space X x Y Another example of a class of sets additive in the weak sense consists of the class of all the sets of an arbitrary metrical space .17, that are both sets ((&<}) and (fto). The smallest class that is additive (in the com- plete sense) and contains these sets is clearly the class of Borel sets in 37. The assertion of (9.7) enables us to prove easily the following theorem due to H. Hahn [2, p. 437J and in some respect analogous to Theorem 6.6: Let Q fee a class of sets, additive in the weak sense in a space T, and let be the smallest class of sets that is additive in the complete sense and contains Q. Sup- pose further that i is a -measure (S) such that the space T either has finite measure (T), or, more generally, is expressible as the sum of a sequence of $cts of finite measure (r). Then (i) for every set E measurable (Z) and for every /0, there ejrists a net FeQ rt , and a set GeQ , such that FC^E^G and that i(E F) e and i(G E)^e; (ii) for every set E measurable (<) there exist a set (Q ( )a) contained in E, and a set (Q<;rt) containing E* which differ from E at most by sets of measure (i) zero. (9.8) Theorem. Let 9 and ty be additive classes of sets in the spaces X and Y respectively, and let // and v be measures defined respectively for these classes. Suppose that fi(X)<^oo and v( K)<oo, or, more generally, that where A*,,e9E, Y n tty, ii(X n )<oo and J'(Y,,)<co for w=-=l, 2, ... Then, for every setQC** Y measurable (9E9)) ? (i) p{E[(x,y)tQ]}, X as function of y, is measurable (?)) in the space I" and v{E[(x,y)eQ]}, y as function of x, is measurable (9E) in the space X; furthermore (ii) ($)f*(E [(x, y)Q]}dv(y) - (9E) v{E [(x, y) * Y x X i; Proof. We may clearly suppose that no two sets A',,, and also no two sets Y n , have common points. The same will then he true of the sets X n x Y m in the space X\ Y. Let us denote by 9? the class of all the sets P measurable (SE^) in the space X x Y, such that conditions (i) and (ii) of the theorem 86 CHAPTER III. Functions of bounded variation. hold good for every set Q=P-(X n x Y m ) where n and m are arbitrary positive integers. Since Q^^Q-(X n x Y m ) for every set (?O*'x 1', n,m and since no two of the sets X n x m have common points, it follows easily from Lebesgue's Theorem 12:3, Chap. I, that every set Q belonging to the class 97 fulfills conditions (i) and (n). We have to prove that this class includes all sets measurable (9t?)). To do this, we observe that it follows at once from the identity (9.1) that every set XxY for which XeS and Ye?), belongs to 31. On the other hand, since by hypothesis n(X tt Xoo and r( F ;i )<oo for every w, we easily deduce from Lebesgue's Theorems 12.3 and 12.11 Chap. I, that the sum of any sequence of sets (97) no two of which have common points, and the limit of any descending sequence of sets (97), are themselves sets (97). The class 97 is therefore normal, and by Lemma 9.4, contains all sets (9E?)). This proves the theorem. If we suppose the hypotheses of Theorem 9.8 satisfied, a meas- ure can be defined for the class 9?) so as to correspond naturally to the measures // and i that are given for the classes 9E and ?). We do this by calling measure (/**) of a set Q measurable ($?)) the common value of the integrals (ii) of Theorem 9.8. It is immediate that we then have pv(Xx Y)=t*(X)'i'(T[) for every pair of sets JCeS and Ye<. This definition enables us to state Theorem 9.8 in a manner analogous to Theorem 8.6. But the analogy would be incomplete if we neglected to extend at the same time the class 9E?). Thus, for instance if 9E and ?) denote respectively the classes of sets meas- urable in the Lebesgue sense in Euclidean spaces Jf p and /f r/ , the class SE?) does not coincide with that of the sets measurable (^) in Kp+q, although it is evidently included in the latter. The extension of the class 9E S 2), that we require in the general case, will lx j defined as follows. Given an additive class of sets Z and a measure T associated with this class, we shall call the class Z complete with respect to the measure i if it includes all subsets of sets (Z) of measure (i) zero. Thus for instance, if /' denotes any measure of Oarathodory, the class IV is complete with respect to F (of. ("hap. II, p. 44), and in particular, the class 8 in a Euclidean space is complete with respect to Lebesgue measure; whereas the class of sets measurable ( S B) is not complete with respect to that measure. 1 9] Fubini's theorem in abstract spaces. 37 Every additive class of sets Z may be completed with respect to any measure t defined for the class, i. e. there is always an ad- ditive class SD$ such that the function of a set T can be continued as a measure on all sets (S) and such that S is complete with respect to the measure i thus continued. Among the classes G of this kind, there is a smallest one that we shall denote by Z*. As is seen di- rectly, this class consists of all sets of the form T N^+N^ where TeZ and N 19 N 2 are arbitrary subsets of sets (Z) of measure (t) zero. The extension of the measure i to all sets of this form is evident. We can now state the following theorem which corresponds to Fubini's Theorem 8.1: (9.10) Theorem. Under the hypotheses_of Theorem 9.8, if f(x,y) is a non-negative function measurable (Wty"") in the space A'x Yj (*i) f( x j y) as function of x is measurable (9E") in X for every yeY, except at most a set of measure (v) zero; (h) K X J y) as function of y is measurable ($") in Y for every xeX, except at most a set of measure (u) zero] (ii) ff(x,y) dnv(x 9 y)= /'[ lf(r,y)dp(x) \ dv(y) = /'[ ff(x,y) di'(y)]dv(x). xx i r- x x i Proof. In the special case in which / is the characteristic function of a set measurable (9Ei)), the theorem is an immediate consequence of Theorem 9.8. The same js^ true when / is the cha- racteristic function of a set measurable (9E?)" 11 ) of measure (fiv) zero, and in consequence Theorem 9.8_remains true when / is the cha- racteristic function of any set (SE?) 1 ' 11 '). This being so, we pass as usual to the case in which / is a finite function, simple and measurable (9E?)'") and finally, with the help of Theorems 7.4 and 12.6, Chap. I, to the^eneral case in which / is any non-negative function measurable (9E 1 ?)" 1 ). Condition (9.9) is essential to the validity of Theorems 9.8 and 9.10. To see this, let us consider some examples for which the condition is not fulfilled. Put X-= l'-/f,, and let 9E= < ?) be the class of all sets in ^ that are measurable in the sense of Lebesgue. We choose for / the ordinary Lebesgue measure, and we define the measure v by making r(F) equal to the number of elements of Y (so that p(r)=oo if Y is an infinite set). Finally, let Q be the set of all the points (x, x) in Ii t =Xx l r such that O^a?^: 1. The integrals occuring in condition (ii) of Theorem 9.8 are then respectively and 1 so that condition (ii) does not hold. (We could also, by a suitable modification of the set Q, choose 1'= ff t and take as measure v the length ^j cf. Chap. II, 8.) 88 CHAPTER III. Functions of bounded variation. Another example showing the importance of condition (9.9) is due to A. Lindenbaurn. Put X= Y=R 19 let =?) be.the class of all Borel sets in JK 19 and let fi(X)t=v(X) denote for every set X the number of its elements. By a theorem of the theory of analytic sets (cf., for instance, C. Kuratowski [I, p. 261]) there exists in the plane /f 2 =ATx V a Borel set Q such that the set of xe jR t for which E [(x, y)*Q] reduces to a single point, is not measurable in the sense of Borel. y In other words, the set of xc X for which *{E[(#, y)Q]} \ is not measurable (SE). y Thus condition (i) of Theorem 9.8 does not hold. For the results of this , vide H. Halm [2]; cf. also S. Ulam [2], Z. Lom- nicki and S. Ulam [1], and W. Feller [1]. For a discussion of Fubini's theorem applied to functions whose values belong to an abstract vector space, vide also S. Bochner [2]. Finally we observe that certain theorems, analogous to those established in this for measurable sets and sets of measure zero, can be stated for the property of Baire and the Baire categories. Cf. on this point C. Ku- ratowski arid S. Ulam [1]. 10. Geometrical definition of the Lebesgue-Stleltjes Integral* The geometrical definition of an integral is inspired by the older and more natural idea of regarding the integral as the measure of an "area", or of a "volume", attached to the function in a certain way that is well known. Let us begin by fixing our notation. Given a function f(x) defined on a set QC_K m , we call graph of f(x) on $, and we denote by B (/;#), the set of all points (x, y) of K m +\ for which x<-Q and y~f(x)^oo. If f(x) is non-negative on Q, the set of all the points (X, y) of R m \-\ such that x*Q and O^y^f(x) is termed, according to C. Oarath<k)dory, ordinate-set of / on Q arid will be denoted by As in 3 7 we shall suppose the space K m fixed and a non- negative additive function U of an interval given in R m . And in accordance with 2, p. 59 and 5, p. 65, L r denotes the Lebesgue measure in JR^ (10.1) Lemma. If QC_R m is a set measurable (2</), the set in R m +\ i$) for every pair of real numbers a and , measurable (2(/L,)> an( l ^ measure (UI^) is (b a) -U*(Q). Proof. Let us write for short, #,/ f a=E[tf0; a^y^b] and T f/L^ We shall begin by showing that if Q has measure (U) zero, the set Q Utb is of measure (T) zero, and so is certainly meas- urable (r) f 10] Geometrical definition of the Lebesgue-Stieltjes integral. 89 To see this, observe that there is then, for any f>0, a sequence of intervals {/} in lt m such that Q(2l n and U(I n )^e> Writing /i n Jn=InX[afj b+e], we obtain a sequence of intervals {J n } in R m +\) such that QatC%Jn and 2T(J n )=2U(I n )-(b a+2c)^(b-a+2e)'f. Thus T*(Q fli6 )=0. Let now Q be any set measurable (#). By Theorem 6.6, Q is the sum of a sequence of closed sets and a set of measure ( U) zero. Therefore, by the above, the set Q n ^ is also the sum of a se- quence of closed sets and of a set of measure (T) zero, and is thus measurable (S^). Finally, for every real number y y we have m[( x >y)eQa.b]Q if 0<y<&, and E[(#,y)6Q, /t6 ]==0 if y is outside the interval [a,6]. Hence, by Theorem 8.6, we have T*(Q atb )= J U*(Q)dy~ a = (b a)-U*(Q), which completes the proof. (10.2) Theorem. If f(x) is a function measurable (fi^/) on a set j its graph on Q is of measure (ULJ zero. Proof. Since any set measurable (Sc/) can be expressed as the sum of a sequence of bounded measurable sets, we can restrict ourselves to the case in which Q is bounded. Let us fix an OO and write Q n =E[xeQ-j ne * for every integer n. By Lemma 10.1 the measure ( ULJ of the graph of f(x) on Q n does not exceed r U*(Q n )> therefore, on the whole setQ, it does not exceed *-U*(Q) 9 and so vanishes. We can now prove the following theorem which includes the geometrical definition of the Lebesgue integral: (10.3) Theorem. In order that a function f(x) defined and non- negative on a set Q(^R m measurable (Zu) be measurable (Zu) on Q, it is necessary and sufficient that its ordinate-set A(/; Q) on Q be meas- urable (fic/ij. When this condition is fulfilledj the definite integral (U) of f on Q is equal to the measure (C/Lj) of the set A(/; Q). Proof. Write, for short, T^U^ and suppose first that f(x) is a simple function, finite, non-negative, and measurable (8i/) on Q, i.e. that /={ 1 ,^ l ; v^Q-^ ; v nj Q n } where Qt are sets measurable (2j/) 90 CHAPTER III. Functions of bounded variation. no two of which have common points. By Lemma 10.1, all the sets A (/;#/) are measurable (i! r ) and T*(A(/; Q t )] =v r U*(Qi) for i=l,2, ..., n. Hence, the set A(/; Q)=A(f; Q t ) is itself meas- t arable (2r), and its measure (T) is equal to %T*[A(f- 9 <?/)] = dU(x). Q Let now / be any non-negative function measurable (fir) on Q. There is a non-decreasing sequence }M#)} of simple functions, finite, non-negative, and measurable (Zu) on<? such We then have (10.4) A(/; (?) - lim /i Now, by the above, all the sets A(A n ; Q) are measurable (fir) and T*[A(h n ;Q)]=fh n dU for n=l,2, ... . On the other hand, by Q Theorem 10.2, the set B(/;$) has measure (T) zero. It therefore follows at once from (10.4) that the set A(/;Q) is itself measur- able (fir) and that - lim Jh n dU=ffdV. It remains to prove that, if the set A(/; Q) is measurable (fir), the function / is measurable (fit/). To do this, write for short ^i=r= A (/;$), and observe that, for every non-negative number y, the set E[#e$; f(x)^y] coincides with the set E[(#, y)eA]. X X Thus, by Theorem 8.6, it A is measurable (fir)i the set E[#eQ? 1( x )^y] X is measurable (%u) for all numbers y except at most those of a set of measure (L x ) zero. But this suffices for the measurability of / on Q (cf. Chap. I, (7.2)) and so completes the proof. * 11. Translations 'of sets. As an application of Theorem 8.6, we shall prove in this a theorem on parallel translations of sets. As a matter of course, in what follows, translations could be replaced by rotations, or by certain other transformations consti- tuting continuous groups and preserving Lebesgue measure. [11] Translations of sets. 91 Given two points x=(x\, # 2 , ..., x m ) and y = (y\, yi, ..., y m ) in the space R m , we shall denote by x-\- y the point (Xi+y\,xi+yi,...,x m +y m ) and by |a?| the number (x\ + x]+ ... +x l m ) 1 \ We shall write #->0 when |o?|~>0. If Q is a set in the space Ii m and a any point of this space, Q (fi) will denote the set of all points x-\-a where x tr Q. The set Q (n) is termed translation of Q by the vector a. If is -an additive function of a set in K m and ae/tf,,,, we shall write ) = 0(X (a) ) for every set X bounded and measurable (33). (11.1) Theorem. If Q is a bounded set (53) of measure (L) zero in the space H rn and & is an additive function of a set (5)) in /,, the func- tion vanishes for almost all translations ofQ, i. e. Q(Q (a) ) = <l> (a \Q)~Q for almost all points a of R m . Proof. We may clearly assume to be a non-negative function, and Q to be a bounded set ((5 d ). Hence, by Theorem 6.10, there is a non-negative additive function U of an interval, such that 0(X)=U*(X) for every set X bounded and measurable (53). Denote by Mj for any set M(^_Ii mj the set of all points (x, y) of the space R^ m which are such that Xf.R m , yelf m and x-\-yfM. The set M is clearly open whenever the set M is open. It follows at once that if -M is a set ( ( s), so is the set M. Finally, observe that for every point zeK m we have E[(#, z) eJlf]=E[(2, y)eM] = M ( * ) - * m u Since the given set Q is, by hypothesis, a set (<*), so is the set#, and by Theorem 8.6, because all translations Q ( ~ z) of the set Q are of measure (!/,) zero. Hence 0(Q(-~*))=U*(Q ( -* ) ) = Q for every zelt nn except at most a set of measure (L m ) zero. Replacing z by a, we obtain the required statement. (11.2) Theorem. Given an additive function of a set <P, each of the following three conditions is both necessary and sufficient for the function to be absolutely continuous: 1 Iim <P(Q (a) ) = Iim (j> (a \Q)=(Q) for every bounded set Q meas- rt-X) <7->0 urable (53) and of measure (L) zero; 2 lim$(Q (fl) ) = Iim0 ( ' l) (0)= <D(Q) for every bounded set Q o-H) ->-0 measurable (35); 3 limW[(P (r/) CP; I]=0 /or eiwy interval I. 92 CHAPTER III. Functions of bounded variation. Proof. It is evidently sufficient to establish the necessity of condition 3 and the sufficiency of condition 1. Suppose first that is an absolutely continuous additive function of a set. In virtue of Theorem 14.11, Chap. I, </> is thus the indefinite integral of a function / measurable (93). Let / = [!, b\\ ...; a m , b m ] be an interval in the space considered and let J be an interval containing I in its interior, for instance the interval [a, 1, 6 t + i; ...; 1, b m +l]. Let e be any positive number. Since the function /(a?) is in- tegrablo overJ, there exists a number i^>0 such that f\f(x) dx<e/3 x for every set JTCV measurable (93) and of measure (L) less than r\. Therefore f\f(x)\dx<e/3 and [\f(x+u)\dx=-- f\f(x) dx<(!3 (H-3) x x (H) if Xe^ XCIj \X\<ri and \u\<\. On the other hand, by Lusin's Theorem 7.1, there exists a closed set FCI such that the function / is continuous on F and such that |I F|<i?/2. Let a<l be a positive number such that JF (U) C/ whenever |w|<a, and such that (11.4) \f(x+u) f(x)\<- whenever x F, x+ucF, and \u\<o. Let now a be any point of K m such that |a|<a. By (11.4) (11.5) On the other hand, \I-F-t* ~ a} \^\IF\ + \IF ( - f ' } \^2.\I-^ and therefore, by (11.3), f\f(x+a) f(x)\dx<2f!3. t- F . F (~n) If we add this inequality to (11.5) we obtain f\f(x+a) -f(x)\ i i. e. the variation W[<P (rt) #; /] =^ '\f( x +a) f(x)\dx tends to / with |a|. The function <P therefore fulfills condition 3. It remains to prove the sufficiency of condition 1. Now, if the function <P fulfills this condition, vanishes by Theorem 11.1 for every bounded set measurable (9)) of measure (L) zero, arid so is absolutely continuous. [12] Absolutely continuous functions of an interval. 93 It was long known that every absolutely continuous function fulfills conditions 1, 2 and 3 of Theorem 11.2. The converse, however, (i. e. the suf- ficiency of these conditions, in order that the function of a set be absolutely continuous) was established more recently. The sufficiency of condition 3 was first proved by A. Plessner [1] (with the help of trigonometric series and for functions of a real variable). As regards the other conditions (1 and 2), and as regards Theorem 11.1, vide II. Milicer-Gruzewska [1], and N. Wiener and R. C. Young [1]. In the text we have followed the method used by the latter authors. 12. Absolutely continuous functions of an Interval. An additive function F of an interval will be termed absolutely con- tinuous on a figure JR , if to each OO there corresponds a number 77>0 such that for every figure R(^R the inequality |-R|<ij implies \F(R)\ < f . In conformity with 3, p. 59, we shall understand by absolute continuity in an open set (7, absolute continuity on every figure RCJQ, and by absolute continuity, absolute continuity in the whole space. Every additive function of an interval, absolutely continuous on a figure J? , is of bounded variation on -R . For, if F is a function that is absolutely continuous on J? , there exists a number 7?>0 such that, for every figure RR , the inequality |JB|<i? implies |.F(/2)|<1. Therefore, if we subdivide R Q into a finite number of intervals /i, /2, > In of measure less than ij, we obtain W(JF; An additive function of an interval F, of bounded variation on a figure J? , will be termed singular on R , if for each e>0 there exists a figure RCR<> such that \R\<* and W(jF; R QR)<e. The reader will observe the analogy between the above definitions and the criteria given in Theorems 13.2 and 13.3, Chap. I, in order that an additive function of a set should be absolutely continuous or singular. This analogy could be pushed further by introducing the notions of absolutely continuous function, and of singular function, with respect to a non-negative additive func- tion of an interval. But this "relativization", although useful in certain cases, would not play an essential part in the remainder of this book. The following theorem is, almost word for word, a duplicate of Theorem 13.1 of Chapter I. 94 CHAPTER III. Functions of bounded variation. (12.1) Theorem. 1 In order that an additive junction of an interval be absolutely continuous [singular] on a figure R , it is neces- sary and sufficient that its two variations, the upper and the lower, should both be so. 2 Every linear combination, with constant coefficients, of two additive functions of an interval which are absolutely continuous [singular] on a figure R Q is itself absolutely continuous [singular] on R . 3 The limit of a bounded monotone sequence of additive functions of an interval that are absolutely continuous [singular] on a figure R Q is also absolutely continuous [singular] on R . 4 // an additive function of an interval is absolutely continuous [singular] on a figure R Q , the function is so on every figure R(2R . 5 // an additive function of an interval is absolutely continuous [singular] on each of the figures R l and R 2 , the function is so on the figure R^R^ 6 An additive function of an interval cannot be both absolutely continuous and singular on a figure R Q , without vanishing identically on R . Part 3, at most, perhaps requires a proof. (It differs slightly from the corresponding part of Theorem 13.1, Chap. I.) Let therefore F be the limit of a bounded monotone sequence {F tl } of additive functions of an interval on a figure R . Let e be any positive number. Since the functions F F n are monotone on R , there exists a pos- itive integer n Q such that (12.2) \F(R)F li{) (R)\^\F(R Q )~F^(R Q )\<el2 for every figure KC^o- This being so, let us suppose that the functions F n are ab- solutely continuous on R Q . There is then an *7>0 such that, for every figure R(2R , |/?|<i? implies the inequality \F n{i (R)\</2 and therefore, by (12.2), the inequality \F(R)\<e. The function^ is thus absolutely continuous on R Q . Suppose next that the functions F H are singular on jK . There is then a figure R^CR* such that l#il<* and w [^i; JK 9 JB i]< c / 2 - Hence, by (12.2), W[F; R QRi]<fj which shows that the function F is singular. This completes the proof. We shall now establish two simple theorems that show ex- plicitly the connection between the absolutely continuous or sing- ular functions of an interval and those of a set. To avoid misunder- standing, we draw the reader's attention to the abbreviations adopted in 5, p. 66, in the terminology of functions of a set. [12] Absolutely continuous functions of an interval. 95 (12.3) Theorem. In order that a non-negative additive function F of an interval be absolutely continuous, it is necessary and sufficient that the eorresponding function of a set F* should be so. Proof. Suppose that the function F is absolutely continuous. In order to prove that the function F* is so too, it is enough to show that F* vanishes on every bounded set of measure (L) zero. Let therefore E be such a set, and let J be an interval that contains E in its interior. For any e > 0, let 17 be a positive number such that (12.4) |JB|<*? implies \F(R)\<.e for every figure RC_J. Since |1|=0, there exists a sequence of intervals {!} in J such that (12.5) Denote by R k the sum of the k first intervals of this sequence. By Theorem 4.6 (or 6.1) of Chap. II, and Theorem 6.2, the relations (12.4) and (12.5) give F*(E)*^ ]im^%R)< lim F(R k )^e, from which it follows that F*(E) = 0. Conversely, if F* is an absolutely continuous function of a set, the absolute continuity of F follows at once from the inequality F(R)^F*(R) which holds by Theorem 6.2 for every figure R. (12.6) Theorem. In order that a non-negative additive function of an interval F be singular ', it is necessary and sufficient that the corresponding function of a set F* should be so. Proof. Suppose that the function of an interval F is singular, and let J be any interval. Given any number f^>0, there is then a figure R(^J such that |J2|<e and F(JQR)<e. Consequently, by Theorem 6.2, we have F*(JR)^F(JQR)<e, which shows on account of Theorem 13.3, Chap. I, that the non-negative function of a set F* is singular in the interior of every interval J, and there- fore in the whole space. Suppose, conversely, that the function of a set F* is singular, and let e be any positive number. Given any interval I there is then a set ECI such that \E\ = and F*(I J5)=0. Consequently, there is a sequence of intervals {!} in I such that (12.7) r EC 2 In and (12.8) Denote by R k the sum of the k first intervals of this sequence. Since |J?|=0, we obtain from (12.7) that |JE/J > \I\ f for a sufficiently large fc , and writing P =1012^, this gives |P|O. Again, by (12.8), F(lQP)<e, which proves that the function F is singular. 96 CHAPTER III. Functions of bounded variation. 13. Functions of a real variable. The most important of the Motions and theorems of this chapter were originally given a rather different form: they were made to refer, not to additive functions of an interval, but to functions of a real variable. It is, however, easy to establish between functions of a real variable and additive functions of a linear interval, a correspondence rendering it immaterial which of these two kinds of functions is considered. To do this, let f(x) be an arbitrary finite function of a real variable on an interval / . Let us term increment of f(x) over any interval I = [a, b] contained in / , the difference f(b) /(a). Thus defined the increment is an additive function of a linear interval /C\r<)> an( i corresponds in a unique manner to the function f(x). Conversely, if we are given any additive function F(I) of a linear interval /, this in itself defines, except for an additive constant, a finite function of a real variable f(x) whose increments on the intervals / coincide with the corresponding values of the function F(I). We shall understand by upper, lower and absolute, variations of a function of a real variable f(x) on an interval I, the upper, lower, and absolute, variations of the increment of f(x) over /. To denote these numbers, we shall use symbols similar, to those adopted for additive functions of an interval, i. e.: W(/; /), W(/; /), and W (/;/). A finite function will be termed of bounded variation on an interval / , if its increment is a function of an interval of bounded variation on / . Similarly the function is absolutely continuous, or singular, if its increment is absolutely continuous, or singular. As we see immediately, in order that a function f(x) be of bounded vari- ation on an interval 7 , it is necessary and sufficient that there exists a finite number M such chat \f(bt) /(a/)|<Jf for every sequence of non-overlapping intervals {[a/, &/]} contained in 7 . Similarly, in order that f(x) be absolutely continuous, it is necessary and sufficient that to each >0 there corresponds an t?>0 such that \f(b { ) /(a/)|<e for every sequence of non-overlapping inter- vals {[a/, bt]} contained in I and for which \bt a i [13] Functions of a real variable. 97 If f(x) and g(x) are two bounded functions on an interval I and M denotes the upper bound of the absolute values of f(x) and g(x) on I , we have \f(b)g(b) f(a)g(a)\ < M[\f(b) f(a)\ + \g(b)g(a)\] for every interval [a, 6]C^o- I* follows at once that (13.1) The product of two functions of bounded variation [absolutely continuous] on an interval is itself of bounded variation [absolutely continuous] on this interval. Finally we see that if a function of an interval F corresponds to a finite function of a point / (i. e. is the increment of /), we have o/(/; 0) = o/(_F; a) for any interval I and any point a el (cf. Chap. II, 3, p. 42, and the present Chapter, 3, p. 60). Thus, in particular, in order that the function / be continuous at a point a according to the definition of 3, Chap. II, it is necessary and sufficient that the function of an interval F that corresponds to / should be so according to the definition of 3 of the present Chapter, If at a point a a function of a real variable / has a unique, limit on the right, this limit will be denoted by /(a+)? similarly, f(a ) will stand for a unique limit on the left. If the function / is defined in a neighbourhood of a point a and both limits /(#+), f(a ) exist, then the oscillation o(/; a) (vide Chap. II, p. 42) is equal to the largest of the three numbers |/(#+) /(a -)|, \f(*+)f(*)\, and |/(a ) /(a)|. If both limits /(&+) and f(a ) exist and are finite, and f(a) =[/(+)+/ (a )] the function f(x) is termed regular at the point a. It is regular if it is regular at every point. Let / be any function of a real variable, of bounded variation, and {a n } a sequence of points. Let us put s(a)~ and ) f(a)+Z (a ' x} [f(a n +)~f(a n )]+f(x)f(x) for x>a for x<a, where the summation (a ' x) is extended to all indices n such that n a<a n <x, when oa, and a>a n >#, when x<a. The function s thus defined is termed the saltus-function of f corresponding to the sequence {a n } of points. It is continuous everywhere except, perhaps, 98 CHAPTER III. Functions of bounded variation. at the points a n , and by subtracting it from / we obtain a function of bounded variation, continuous at all points of continuity of / and, besides, at all the points a n . If {a n } is the sequence of all points of discontinuity of /, the corresponding function s is called simply the saltus-function of f. By varying the fixed point a we get the various saltus-functions of / which can obviously differ only by constants. A function of bounded variation which is its own saltus- function, is called a saltus-function. The functions of a real variable whose increments over each interval I coincide respectively with the variations W(/; I), W(/; I) and W(/; I) of a function /, are also termed (upper, lower, and ab- solute) variations of f. By applying the Jordan decomposition ( 4, p. 62), we can express any function of a real variable / of bounded variation as the sum of two functions that are respectively its upper and lower variations. Thug any function of bounded variation is the difference of two monotone non-decreasing functions, and con- sequently is measurable (93) and has at every point the two uni- lateral limits, on the right and left. Moreover, the set of its points of discontinuity is at most enumerable, since the sum of its oscil- lations at the points of discontinuity lying in any finite interval is always finite (this is actually the special case of Theorem 4.1). In various cases it is more convenient to operate on functions of a real variable than on additive functions of an interval in U^ The difference is, of course, only formal, and all the definitions adopted for functions of an interval can be stated, with obvious modifications, in terms of functions of a real variable. We need not state them here explicitly. If F is a function of a real variable, of bounded variation, the meaning of expressions such as Lebes- gue-Stieltjes integral with respect to F, integral (F), sets (Sf), and so on, may be regarded as absolutely clear, in view of the definitions of 5. If F is a continuous function and g a function integrable (F), the integral fgdF, where 1 is a variable / interval, is an additive continuous function of an interval / (vide 5, p. 65). There is, consequently, a continuous function of a real variable whose increment on any interval I coincides with the definite integral (F) of g over I. This function, which is determined uniquely except for an additive constant, is also termed indefinite integral (F) of g. [13] Functions of a real variable. 99 When there is no ambiguity, the additive function of an interval that is determined by a finite function of a real variable F, will be denoted by the same letter F, L e. F(I) will stand for the in- crement of F(x) on an interval I. By means of the corresponding function of an interval, any function of a real variable F of bounded variation determines an additive function of a set which we denote by F* (cf. 5). We see at once that F*(X )=F(b+) F(a ) when X=[a, ft], and that F*(X)=F(a+) F(a ) when -T=(a), i. e. when X is the set consisting of a single point a. If W(x) is the absolute variation of a function F(x) of bounded variation, we clearly have W(JP*; J^X'W r *(Z) for every set X bounded and measurable (53). The opposite inequality does not hold in general. If, for instance, X is a set consisting of one point only, and F is the characteristic function of X, then W(F*; X)= =JF*(.C)==0, while W*(X)=2. We can, however, state the following theorem: (13.2) Theorem. If F(x) is a function of a real variable of bounded variation, and W(x) is the absolute variation of F(x), then W(jF*; X)= = W*(X) for every set X bounded and measurable (93) at all points of which F(x) is continuous. Proof. Suppose first that the set X is contained in an open interval J i* 1 which the function F(x), and consequently the function W(x) also, is continuous. Let (?C<A> ^ e an arbitrary open set such that XQ. Then, expressing as the sum of a sequence of closed non-overlapping intervals {!}, we get *-,In)=Vr(F*-,G)i whence W*(ZKW(F*;Z), and /i /i since the opposite inequality is obvious, W*(X) = W(F*; X). Let us pass now to the general case. Let 7 be an interval containing X in its interior, and let >0. Denote by {a,,} the se- quence of points of discontinuity of F(x) interior to I , and by SN(X) the saltus-f unction of F(x) corresponding to the points a n for n>N. Let us put 0(x)=F(x) SN(X), where N is a positive integer suf- ficiently large in order that W(flW; I )^ e - The points aj, a 2 , ..., a^, none of which belongs to X , divide 7 into a finite number of sub- intervals J*,J\,...,JN in the interior of which the function 0(x) is continuous. Hence, denoting by V(x) the absolute variation of 100 CHAPTER III. Functions of bounded variation. there follows, by what has already been proved, V*(X-J k ) = X.J*) for fc = 0, 1, ..., N, whence F*(X)=W(<?*; X). On the other hand, \W*(X)V*(X)\ and |W(JP*; X) W((?*; X) are both at most equaito W(flW; Io)<*. Thus \W*(X)Vr(F*;X)\^2t, and finally W*(X) = W(F*; X). If F(J?) is a finite function of a real variable and E an arbit- rary set" in /?!, the set of the values of F(x) for yeE will be denoted by F[E]. (13.3) Theorem. If F(x) is a function of a real variable of bounded variation and W(x) is the absolute variation of F(x), then \F[E]\^W*(E) for every set E in JB^ and if further the function F(x) is non-decreasing, and continuous at all points of J3, then Proof. Let e be a positive number and {/} a sequence of intervals such that #ClX and W*(E)+e^W(I n ). Then, if m n and n n M n denote the lower and upper bounds, respectively, of F(x) on /, the sequence of intervals {[w, M n ]} covers the set F[E]j and con- sequently \F[E]\^Z(M n m n )^ZW(In)^W*(E)+c. Hence, Suppose now F(x) continuous at the points of E and non- decreasing. By what has already been proved, |jP[ t ]|^ J P*(jE?). To establish the opposite inequality, let r\ be an arbitrary positive number, and { J n } a sequence of intervals subject to the conditions F[E]C%Jn. and |^S]|+i?>IiVl- Let E n denote the set of the n n points xeE such that F(x)eJ n . Then F*(E n )^\J n \ for each n; and consequently F*(E)^2\J n \^\F[E]\ + vi, whence F*(E)^\F[E]\. n The characteristic function of a set consisting of a single point provides the simplest example of a singular function of a real variable, that does not vr.".ish identically. This function is however discontinuous. It is easy to give examples of functions of an interval that are additive, singular, continuous, and not iden- tically zero, in the spaces Jt m for ra^2. For simplicity, consider the plane, and denote, for any interval /, by F(I) the length of the segment of the line y = x contained in I; the function of an interval .F(I) will evidently have the desired properties. A similar example for JK^ is less trivial. We shall therefore conclude this with a short description of an elementary method for the construction of continuous singular functions of a real variable. We shall begin with the following remark which frequently proves useful. [13] Functions of a real variable. 101 (13.4) Let E be a linear, bounded, perfect and non-dense set, and a and ; a two arbitrary numbers. Then, if a and b denote the lower and upper bounds of E 9 a function F(x) may be defined on the interval J [a, b] so as to satisfy the following conditions: (i) F(a) = a, F(b) = /?, (ii) F(x) is constant on each interval contiguous to the set E, and (iii) F(x) is continuous and non-decreasing on the interval J and strictly increasing on the set E. To see this, let {l n } be the sequence of intervals contiguous to E, and let us agree to write In^Im whenever the interval I n is situated on the left of I m . By induction (cf. e. g. F. Hausdorff [II, p. 50]) we can easily establish a one- to-one correspondence between the intervals I n and the rational numbers of the open interval (a, ft) so that, denoting by u(I n ) the number which corresponds to the interval I n , the relation 7/,^/m implies u(I n ) u(l m ). Let us now put F(x)=u(I n ) for xel n where n 1, 2, ..., and then extend F(x) by continuity to the whole of the interval t/ . We see at once that the function F(x) thus ob- tained satisfies alJ the required conditions (i), (ii) and (iii) of (13.4). Now let us choose for the set E in (13.4) a set of measure zero. Then if {In } is the sequence of the intervals contiguous to E, we have \ A- 1 n for each positive integer n; and since |J lk\ >0 as n->oo, the func- tion F(x) is evidently singular on the interval J . The singular function obtained by the foregoing construction is continuous and monotone non -decreasing; the function is not constant on the whole interval J , but is so on certain partial intervals. Now, by the method of condensation of singularities, it is easy to derive from it a singular continuous function that increases everywhere. To do this, suppose in (13.4), \E\ 0, a= a= 0, b = fl 1, and extend the function F(x) on to the whole axis R v by stipulating F(x+ 1)= l+F(x). Write (13.5) This series is a uniformly convergent series of singular functions, since F(iur) is clearly singular with F(x). Now the functions F(nx) are monotone non -decreasing. By Theorem 12.1 (3), the function H(x) is thus singular. This function is also con- tinuous, as the limit of a uniformly convergent aeries. To prove that H (x) is strictly increasing, let x l and x l >x l denote an arbitrary pair of points in [0, 1], For n >!/(, xj, we have HX, ft#|>l, and consequently F(nx t )>F(nx l ); while for every n, Finx^^Finx^ whence by (13.5), H(x t )>H(x 1 ) as asserted. Various examples of this kind have been constructed by A. Den joy [l] t W. SierpiAski [3], H. Hahn [I, p. 538], L. C. Young [1] and G. Vitali [4]; cf. also 0. D. Kellog [1], and E. Hille and J. D. Tamarkin [1]. 102 CHAPTER III. Functions of bounded variation. 14. Integration by parts. As in the preceding , we shall deal only in this with functions of a real variable. For the latter, we shall establish two classical theorems, of importance on account of their many applications to various branches of Analysis. We shall first prove them for the Lebesgue-Stieltjes integral and then spe- cialize them for the ordinary Lebesgue integral. (14.1) Theorem on integration by parts. If U(x) and V(x) are two functions of bounded variation, we have for every interval I = (a,b] b ft f'UdV + fVdU=U(b+)V(b+) U(a)V(a-~), a (i prodded that at each point of / either one at least of the functions U and V is continuous, or both are regular. Proof. In order to simplify the notation assume a = and &= 1, and consider the triangle Q E[0^#^l; y^x] on the plane JB 2 . The set E[(J?, y)eQ] is then the interval [y, 1] or the empty set, X according as y belongs, or does not belong, to the interval [0, 1]. Similarly, E [(,r, y)eQ] is the interval [0, x] or the empty set, y according as we have, or do not have, OO<;i. Hence, by Fubini's theorem in the form (8.6), i. e. i (14.2) o Interchanging U and V and adding the corresponding equation to (14.2), we get, on dividing by 2, i i (14.3) f [U(x+)+U(x )]dV(x) + ft[V(x + )+V(x )]dU(x) = 6 o = J7(1 + )F(1 + ) #(0 )F(0 ). Let M be the set of the points in /V=[0, 1] at which the function U(x) is regular. Then (14.4) [14] Integration by parts. 103 On the other hand, the set / M is at most enumerable and, by hypothesis, the function F(#), and consequently both its relative variations, are continuous at each point of I Q M. Thus, the def- inite integral ( V) of any function over the set 1 Q M is /ero, and i i it follows from (14.4) that f^[U(x+)+U(x )]dV(x)=fU(x) dV(x). Similarly, the second member on the left-hand side of the rela- i tion (14.3) is equal to jV(x)dU(x), and this relation may be written o i fUdV+ fVdU=U(l + ) F(l+) f7(0 ) F(0 ), which proves the theorem. The theorem may be also proved independently of Fubini's theorem, but then the proof is slightly longer. The proof given abo\e was communicated to the author by L. C. Young. (14.5) Second Mean Value Theorem. If U(x) and V(x) are two non-decreasing junctions and the function V(x) is continuous, then in any interval [a, b] there exists a point such thot (14.6) fUdV=U(a).(V(S)V(a)]+U(b)-[V(b)-V()l a Proof. Since the values of U(x) outside the interval [a, b] do not affect (14.6), we may suppose that U(a )U(a) and, U(b-\-)=U(b). Therefore, making use of Theorem 14.1 and of the first mean value theorem (Chap. I, Th. 11.13), we obtain (14.7) - 17(6) V(b) U(a) V(a) j* [ 17(6) Z7(a)], where p is a number lying between the bounds of the function V(x) on [a, 6]. But, since this function is by hypothesis continuous, there exists in [a, 6] a point g such that j<=F(). Substituting this value for n in (14.7) we obtain the relation (14.6). 104 CHAPTER III. Functions of bounded variation. As a special case of Theorem 14.1 we have the following theorem on integration by parts for the Lebesgue integral: (14.8) Theorem. If u(x) and v(x) are two summable functions on an interval [a, b] and U (x) and V (x) are their indefinite in- tegrals (L), then b b (14.9) fU(x)v(x)dx+fV(x)u(x)dx^U(b).V(b)U(a)-V(a). b a Proof. Observe first that by writing for instance u(x) = and v(x) = outside the interval [a, 6], we may suppose that the functions u(x) and v(x), and their indefinite integrals U(x) and V(x), are defined on the whole straight line jR x . Also, by altering, if neces- sary, the values of the functions u(x) and v(x) on a set of measure (L) zero, which does not affect the values of the integrals in (14.9), we may suppose that these functions, together with the functions U(x)v(x) and V(x) ii(x), are measurable (^) (cf. Theorem 7.6 of Vitali-Carath^odory or else Lusin's Theorem 7.1). We may, therefore write, according to Theorem 15.1, Chap. I, b b b b fV(x)v(x)dx=fU(x)dV(x) and fv(x)u(x)dx= f'V(x)dU(x), an a a and (14.9) follows at once from Theorem 14.1. Similarly, we derive at once from Theorem 14.5 the second mean value theorem for the Lebesgue integral: (14.10) Theorem. If U(x) is a non- decreasing function on an interval [a, b] and v(x) is a summable function on this interval, then b | b J V(x) v(x) dx = V(a) fv(x) dx + U(b) fv(x) dx, a ii where is a point of [a, b]. CHAPTEE IV. Derivation of additive functions of a set and of an interval. 1. Introduction. In this chapter we shall study Lebesgue's theory of derivation of additive functions in a Euclidean space of any number of dimensions. When other spaces are considered, or when we specialize our space (to be, say, the straight line J^ or the plane J? 2 )> we shall say so explicitly. In what follows, an essential part is played by Vitali's Covering Theorem (vide, below, 3) which is restricted to the case of Lebes- gue measure. For this reason, the theorems of the present chapter have not in general any complete or direct extension to other meas- ures, not even when the latter are determined by additive functions of an interval. In accordance with the conventions of 5, Chap. Ill, the terms measure, integral, almost everywhere, etc. will be understood in the Lebesgue sense whenever we do not explicitly assign another meaning to them. Similarly, by additive functions of a set we shall always mean functions of a set (93) (some of which may of course be continued on to wider classes of sets, cf. Chap. Ill, 5). We have already remarked in 1, Chap. I, that any additive function of a set 9 in a space R mj may be regarded as a distribution of mass. It is then natural to consider the limit of the ratio <D(S)I\S\ where 8 denotes a cube, or a sphere, with a fixed centre a and with diameter tending to 0, as the density of the mass at the point a. By the fundamental theorem of Lebesgue (vide, below, Theorem 5.4) this limit exists almost everywhere. Moreover Lebesgue has shown that in the above ratio, 8 may be taken to denote much more gene- ral sets than cubes or spheres. Of these, further details will be given in the next . 106 CHAPTER IV. Derivation of additive functions. 2. Derivates of functions of a set and of an interval* Suppose given a Euclidean space /?,. By parameter of regularity r(E) of a set E lying in this space, we shall mean the lower bound of the numbers \E\I\J\ where J denotes any cube containing . Thus when E is an interval, l m /L' n *^r(E) ^l/L, where I denotes the smallest and L the largest of the edges of E; in particular, the parameter of regularity of a cube is equal to 1. A sequence of sets {E n } will be termed regular, if there exists a positive number a such that r(E n )>a for n=l, 2, ... . We shall say that a sequence of sets {E n } tends to a point a, if ^( ,,)->0 as n->oo, and the point a belongs to all the sets of the sequence. Given a function of a set $ (not necessarily additive) we call general upper derivate of $ at a point a the upper bound of the numbers I such that there exists a regular sequence of closed sets {E n } tending to a, for which lim <P(E n )/\E n \ = l. We shall denote this n derivate by D$(a). Similarly, merely replacing the closed sets by intervals, we define the ordinary upper derivate of at a point a, and we denote it by $(a). If we remove the condition of regularity of the sequences of intervals considered, we obtain the definition of strong upper derivate. In other words the strong upper derivate of $ at a point a is the upper limit of the ratio <P(Z)//i, where / is any interval containing a, whose diameter tends to zero. This derivate will be denoted by $(a). The three lower derivates at a point a, general D0(a), ordinary 0(a), and strong $ s (), have corresponding definitions, and if at a point a the numbers D$(a) and D$(a) are equal, their common value is termed general derivative of <P at the point a and will be denoted by D#(a). If further D#(a)-^oo, the function $ is said to be derivable in the general sense at the point a. Similarly we define the ordinary derivative $'(a) and the strong derivative $' 8 (a), as well as derivability in the ordinary sense, and in the strong sense, of the function at the point a. Sometimes the derivatives D#(a), '(a) and ( ^ s () are termed unique derivates, while the upper and lower derivates (general, ordinary and strong) are termed extreme derivates. At any point a, we clearly have D<P(aX#(aX #(aXD$(a) and similarly s (a) ^$(a) <I<P(a) ^ <P 8 (a); so that the existence either of a general derivative, or of a strong derivative, always implies that of an ordinary derivative. On the other hand, no such relation holds between the general and the strong extreme derivates. [ 2] Derivates of functions of a set and of an interval. 107 It may be noted that in order that it be possible to determine the general derivates of a function of a set <P, the latter must be defined at any rate for all closed sets; whereas in order to determine the extreme ordinary derivates, or the extreme strong derivates, we need only have the function (P defined for the intervals. This is why the process of general derivation is most frequently applied to additive functions of a set, and that of ordinary, or of strong, derivation to functions of an interval. We shall often omit the terms "ordinary", "in the ordinary sense", in expressions such as "ordinary derivate", "derivability in the ordinary sense". We have seen (Chap. Ill, 5) that an additive function of an interval F of bounded variation determines an additive function of a set F*. Let us mention, in the case in which the function F is non-negative, an almost evident relation between ordinary deri- vates of F and general derivates of F*: (2.1) Theorem. If F is an additive non-negative function of an inter- val, then, at any point x, which belongs to no hyperplane of discontinuity of F, we have I)F*(x)^F(x)^F(x)^DF*(x). In particular therefore, the ordinary derivative F'(x)~DF*(x) exists at almost every point x at which F* has a general derivative. Proof. Since F(I)^.F*(I) for any interval I, the inequality F(x)^DF*(x) is obvious. On the other hand, let I denote any number exceeding F(x). Then there exists a regular sequence of intervals {!} tending to the point x and such that lim jF(I l ,)/|Z JI |<L Since x does n not belong to any plane of discontinuity of F, we may asssume that it is an internal point of all the intervals /. Hence we can make correspond to each interval I n an interval J n dln such that xcJ n , |J n |Xl l/n)|Ifl| and r(J /l ) = r(I /l ). We then have Now since {J n } is a regular sequence of intervals tending to x, it follows that T)F*(x)^.l, and therefore that I>F*(x)^F(x). Let us note also the following result: (2.2) Theorem. If f is a summable function and /> is the indefinite integral off, then D0(0X/(#) and a (x) ^f(x) at any point x at which the function f is upper semi-continuous, and similarly D(P (#)>/(#) and Q>(x)^f(x) at any point x at which the function f is lower semi-continuous. In particular therefore, <D'(x)= 08(o?) = D#(#)== f(x) at any point x at which the function f is continuous. 108 CHAPTER IV. Derivation of additive functions. In 1?! there is no difference between ordinary and strong deri- vation. If F(x) is a finite function of a real variable, we understand by its extreme derivates F(x) and F(x), and by its derivative, or unique derivate, F'(x), the corresponding derivates of the function of an interval that F(x) determines (cf. Chap. Ill, 13). Besides these derivates, which we shall often term bilateral, we also define, for functions of a real variable, unilateral derivatives and derivates. Thus, if F(x) is a finite function of a real variable defined in the neigh- bourhood of a point x , the upper limit of [F(x) F(x )]/(x x ) as x tends to X Q by values of #># is called right-hand upper derivate of the function F at the point x and is denoted by F^(x Q ). Similarly we define at the point x the right-hand lower derivate F + (x ) and the two left-hand, upper and lower, derivates, F~ (x ) and F~(x ). These four derivates are called unilateral extreme, or Dini, derivates. If the two derivates on one side (right or left) are equal, their common value is called unilateral (right-hand or left-hand) derivative of the function F at the point in question. Finally, we shall call inter- mediate derivate of F(x) at the point x , any number I such that there exists a sequence {#} of points distinct from X Q for which n =x Q and \im[F(x n )F(x )]l(x n x Q ) = l. Let E be a linear set, x a point of accumulation of E, and F(x) a finite function defined on E and at the point x . The upper and lower limit of the ratio [F(x) F(x Q )]/(x # ) as x tends too? by values belonging to the set E, are called respectively the upper and lower derivate of F at X Q , relative to the set E. We shall denote them respectively by F E (x Q ) and F E (x Q ). When they are equal, their common value is termed derivative of F at X Q relative to the set E, and is denoted by F'E(X O ). Besides this derivation relative to a set, we define also derivation relative to a function. Suppose given two finite functions F(x) and U(x), and let x be a point such that the func- tion U is not identically constant in any interval containing o? . We then call upper derivate FU(XQ) and lower derivate FU(O) of the function F with respect to the function U at the point x , the upper limit and the lower limit of the ratio [F(x) F(x Q )]/[U(x) U(x )] as x tends to x by values other than those for which F(x) F(x )= U(x) U(x )=Q. Similarly, considering unilateral limits of the same ratio, we define four Dini derivates of F with respect to U: [ 3] Vitali's Covering Theorem. 109 When all these extreme derivates are equal, their common value is denoted by F'U(XQ) and called derivative of F with respect to the function U at the point X Q . The most usual case in which this method of derivation is applied, is when U is a monotone increasing function; and it is then easy, by change of variable, to reduce deri- vation with respect to U to ordinary derivation. 3. Vitali's Covering Theorem. We shall say that a family <E of sets covers a set A in the sense of Vitali, if for every point x of A there exists a regular sequence of sets (() tending to x (cf. p. 106). (3.1) VitaWs Covering Theorem. If in the space JK m a family of closed sets ( covers in the sense of Vitali a set A, then there exists in (E a finite or enumerable sequence (E n ] of sets no two of which have common points, such that (3.2) \A ZJ57 n |=0. /} Proof, a) We first prove the theorem in the special case in which (i) the parameters of regularity of all the sets (() exceed a fixed number a>0 and (ii) the set A is bounded i. e. contained in an open sphere S. We may clearly assume that, in addition, all the sets (() are also contained in 8. This being so, we shall define the required sequence {E n } by induction in the following manner. For E l we choose an arbitrary set ((), and when the first p sets E!, EM ... , E p no two of which have common points, have been defined, we denote by 6 P the upper bound of the diameters p^ of all the sets (() which have no points in common with EI, and /i by E p+ \ any one of these sets with diameter exceeding 6 p /2. Such a set must exist, unless the sets E 19 JB a , ... , E p already cover the whole of the set J., in which case they constitute the sequence whose existence was to be established. We may therefore suppose that this induction can be continued indefinitely. To show that the infinite sequence {E n } thus defined covers A almost entirely, let us write (3.3) B=A and suppose, if possible, that |JB|>0. On account of condition (i), we can associate with each set E n a cube J n such that E n ^Jn and 110 CHAPTER IV. Derivation of additive functions. |Jn|>'|t7n|. Let Jn denote the cube with the same centre as J n and with diameter (4m+l)d(J). The series (3.4) ^ converges; therefore we can find an integer N such that ^ \J n \<\B\. n--;V+l ^ It follows that there exists a point x B not belonging to any J n for n>N; and since by (3.3) the point X Q does not belong to and the sets E n are supposed closed, there must exist in ( a set E containing x and such that (3.5) E-En^Q for n = 1, 2, ... , N. Hence the set E has common points with at least one of the sets E n for w>J\ r ; for otherwise we should have < d(E)^ti n ^2d(E n +i)^ : <; 2($(Jn+i) for every positive integer n, and this is clearly impossible since by (3.4) we have lim 4(J n ) = 0. Let n be the smallest of the values of n for which E-E n ^O. Then on the one hand, E-E n = for w = l, 2, ... , tt 1, so that (3.6) -! ; and on the other hand, by (3.5), n Q >N, which implies, by definition of x QJ that XQ does not belong to J ny . Thus we find that there are both some points outside Jn o and some points belonging to the set Er* dJn i which are contained in the set jB; this set must therefore have diameter exceeding 2<5(Jn )^2<J(fin )><5n -i, in contradiction to (3.6). The assumption that |JB| > thus leads to a contradiction and this proves the theorem, subject to the additional hypotheses (i) and (ii). b) Now let ( be any family of closed sets covering the set A in the sense of Vitali; and let us denote, for any positive integer n, by S n the sphere 8(0; n) and by A n the set of the points xeA-S n for which there exists a sequence of sets tending to x and consisting of sets (() whose parameters of regularity exceed 1/n. The sets A n constitute an ascending sequence and jL=lim^ln. n We can now define by induction a sequence of families of sets {Zn} subject to the following conditions: 1 each family Z n consists of a finite number of sets (() no two of which have common [3] Vitali's Covering Theorem. Ill points and none of which, for n>l, have points in common with the sets of preceding families 3^, $ 2 , ... , S-i; 2 denoting by T n the sum of (he sets which belong to Z n , (3.7) \A n To see this, suppose that for n^p we have determined families Z n * subject to 1 and 2. Write A p +\ A p +\ 2j^h an d consider the ti family of all the sets (() which are contained in the open set Cj^T/ and whose parameters of regularity exceed l/(p + l). This family evid- ently covers the set A p +}(2Ap+\(2 Sp+i in the sense of Vi tali, and by what we have already proved we can extract from it a sequence {E { } of sets, no two of which have common points, so as to cover A p +\ almost entirely. Therefore, for a sufficiently large index i , and, denoting by Z p + \ the family consisting of the sets E 19 E 2J ... , E iQj we find that conditions 1 and 2 hold when n p + I. Let us now write Z= Z n . The family Z consists of a finite /I number, or of an enumerable infinity, of sets (() no two of which have common points. Denoting the sum of these sets by T, we find, by (3.7), \A T| = and this completes the proof. The proof given above is due to S. Banach [3] (for other proofs cf. C. Ca- ratheodory [II, pp. 299307] and T. Radri [3]). Theorem 3.1 was proved by G. Vitali [3] in a slightly less general form; he assumed the family (E to consist of cubes. H. Le- besgue [5] while retaining the line of argument of Vitali, showed that the conclusion drawn by Vitali could be generalized as follows: (3.8) Theorem. Given a set A and a family ( of closed sets, suppose that with each point xeA we can associate a number a>0, a sequence {Xi} of sets (2) and a sequence {J n } of cubes such that o?eJ/,, X,C<A* |Xi|>a-|J M | for n=l,2,..., and lim(5j(Jn)==0. n Then 'i contains a sequence of sets no two of which have common points, that covers the set A almost entirely. 112 CHAPTER IV. Derivation of additive functions. This statement, although apparently more general than that of Theorem 3.1, easily reduces to the latter. For let us denote by ( the family of all the sets of the form E + (x) 9 where E is any set of G and x any point of the set A. The family (g clearly covers the set A in the Vitali sense, and by Theorem 3.1 we can therefore extract from it a sequence {En} of sets no two of which have common points And which covers the set A almost entirely. Now each set En either already belongs to (E, or becomes a set (ffi) as soon as we remove from it a suitably chosen point. Therefore, removing where necessary a point from each En, we obtain a sequence of sets (() no two of which have common points and whose sum, since it differs from En by an at most enumerable set, covers A almost entirely. n For further generalizations of Vitali's theorem (which, again, can be proved without introducing fresh methods), see B. Jessen,. J. Marcinkiewicz and A. Zygmund [1, p. 224]. It is easy to see that the hypothesis that the family ( covers the set A in the Vitali sense (and not merely in the ordinary sense) is essential for the validity of Theorem 3.1. But, as has been shown by S. Banach [1] and H. Bohr (vide C. Carath^odory [II, p. 689]), this hypothesis cannot be dispensed with in the theorem even in the case where (E is a family of intervals such that to each point x of the set A there corresponds a sequence {/ n } of intervals belonging to (, of centre x and diameter <$(!) tending to zero as n >oo. For covering theorems similar to that of Vitali and which correspond to linear measure (length, cf. Chap. II, 9) of sets, vide W. Sierpinski [7], A. S. Besicovitch [1] and J. Gillis [1]. $ 4. Theorems on measurability of derivates. Of the two theorems which we shall establish in this , the first is due to 8. Banach [4, p. 174] (cf. also A. J. Ward [2, p. 177]) and concerns the extreme derivates of any function of an interval (not necessarily additive). We begin by proving the following lemma: (4.1) Lemma. Any set expressible as the sum of a family of intervals is measurable. Proof. Let 3 be any family of intervals and let 8 be the sum of the intervals of 3 Let ( denote the family of cubes each of which is contained in one at least of the intervals (3). The set 8 is clearly covered by ( in the Vitali sense and by Theorem 3.1, 8 is therefore expressible as the sum of a sequence of cubes (() and of a set of measure zero. Therefore the set 8 is measurable, as asserted. (4.2) Theorem. IfFis a function of cm interval^ its two extreme ordinary derivates F and F and its extreme strong derivates F B and F n are measurable. [ 4] Theorems on measurability of derivates. 113 Proof. Let us take first the strong derivates of F, say F s . Let a be a finite number and P the set of the points x for which FS(X)>O>- For any pair of positive integers h and fc, let us denote by P/I.A the sum of all the intervals I for which <5(/)^l/fc and F(I)/I\^a + li/h. We see at once that P=^[JPh^ Now the sets /i A P h , k are measurable on account of Lemma 4.1 and so is the set P. This proves f\(x) to be a measurable function. Consider next the ordinary upper derivate F. As before let a be any finite number, and Q the set of the points x at which F(x)>a. In order that a point x should belong to Q, it is clearly neces- sary and sufficient that there should exist a positive number a and a sequence {!} of intervals tending to x such that r(Zn)^a and F(l n )l\ln\^ a + a for n = l,2, .... Hence denoting for any pair of positive integers h,k by $/,,* the sum of the intervals / such that r(I)^l/A, 6(1) ^Ijk and F(I)/\1\ ^ a+l/A, we find easily that Q=2jllQhk. Thus, since each set $/,* is measurable by Lemma 4.1. h h _ so is also the set Q. The derivate ^(^) is therefore measurable. It follows in particular from Theorem 4.2 that the bilateral extreme derivates of any function of a real variable are measurable. The same is not true of unilateral extreme derivates. Neverthe- less, as shown by S. Banach [2] (cf. also H. Auerbach [1]), these derivates are measurable whenever the given function is so. Similarly by a theorem of W. Sierpiriski [8], the Dini derivates of a function measurable ($) are themselves measurable (>). These two results are included in the following proposition, from which they are obtained by choosing the class 9E to be either or "B. (4.3) Theorem. If 96 is an additive class of sets in ff^ which includes the sets measurable ( S B), the Dini derirates of any function of a real variable which is finite and measurable (9E), arc themselves measurable (9E). Proof. If F is any finite function of a real variable, x any point and /*, k any pair of positive integers subject to >/?, let u.s write D/ ltk (F^x) for the upper bound of the ratio [F(t) /'"MIU <v) when x + llk<t<jr + l/h. At any point ,/ we clearly have (4.4) Now let a be any finite number and consider the set (4.5) E[D,,. ft (F;x)>a]. 114 CHAPTER IV. Derivation of additive functions. We see at once that if the function F is constant on a set 15, the set of the points x of E at which D h ,k(F\ x)>a is open in E (cf. Chap. II, p. 41). Thus the set (4.5), and consequently the expression D h ,k(F',x) as function of a?, is measurable (9E) whenever the function F is finite, measurable (9E) and assumes at most an enumerable infinity of distinct values. This being so, let F be any finite function measurable (9C). We can represent it as the limit of a uniformly convergent sequence {F n } of functions measurable (9E) each of which assumes at most an enumerable infinity of distinct values: for instance we may write F n (x) = iln, when i/n^F(x)<(i+l)/n for i=... 2, 1,0,1,2,.... We then have Dh,k(F\x) =]imD^it(F n ]x)^ and since by the above n the functions Dh, k (F n ;x) are measurable (9C) injr, so is D h ,k(F',x). It follows at once from (4.4) that the derivate F+(x) is also meas- urable (9C), and this completes the proof. 5. Lebesgue's Theorem. We shall establish in this the fundamental theorem .of Lebesgue on derivation of additive functions of a set and of additive functions of bounded variation of an interval. (5.1) Lemma. If for a non-negative additive function of a set <& the inequality D0(x)^a holds at every point a; of a set A, then (5.2) holds for every set X~^A, bounded and measurable (33). Proof. Let e be any positive number and b any number less than a. By Theorems 6.5 and 6.10 of Chap. Ill, there exists a bounded open set G such that (5.3) XCG and Let us denote by ( the family of closed sets E(^G for each of which $(E)^b-\E\. Since by hypothesis, T)<I>(x)^a>b at any point xt- A, the faynily ( covers the set A in the Vitali sense, and by Theorem 3.1, we can extract from it a sequence {E n } of sets no two of which have common points, so as to cover almost entirely the set A. Therefore, on account of (5.3), In this we make *->0 and b >a, and (5.2) then follows at once. [ 5] Lebesgue'a Theorem. 115 (5.4) Lebesgue's Theorem. An additive function of a set is almost everywhere derivable in the general sense. An additive function of bounded variation of an interval is almost everywhere derivable in the ordinary sense. Proof. On account of Theorem 2.1 we may restrict ourselves to non-negative additive functions of a set. __ Let be such a function and suppose that D<P(#)>I)<P(#) holds at each point a? of a set A of positive measure. For any pair of positive integers h and k let us denote by A h ,k the set of the points x of A for which I)$(x)>(h+l)lk>hlk>I)$(x). Wehave 4 = &*,*, h, k and therefore there exists a pair of integers h Q and & such that |^.A ,*J>0. Let us denote by B any bounded subset of -A /,,* of positive outer measure. Let be any positive number and a bounded open set such that (5.5) BCO and \0\^\B\ + e. Consider the family of all closed sets ECO for which 0(E)^^-\E\. K Q This family covers the set B in the Vitali sense, and therefore con- tains a sequence (E n ] of sets no two of which have common points, which covers the set B almost entirely. Writing Q = E n we find, on n account of (5.5), (5.6) On the other hand, D0(a?)>(A +l)/* at each point x* B. Therefore, since all but a subset of measure zero of the set B is contained in the set #, it follows from Lemma 5.1 that 0(Q)^-^~--\B\ y and therefore, KQ on account of (5.6) that (A +l)-||<A -(l J5 l + e )- But this is clearly a contradiction since |JB| > and e is an arbitrary positive number. Thus the function 9 has almost everywhere a general derivative D$. It remains to prove that the latter is almost everywhere finite. Suppose the contrary: there would then exist a sphere 8 such that D$(x)=+oo at any point # of a subset of 8 of positive measure. We should then have, by Lemma 5.1, #()=+ oo, which is impos- sible and completes the proof. 116 CHAPTER IV. Derivation of additive functions. The preceding theorem was proved by H. Lebesgue [I, p. 128] first for continuous functions of a real variable and later [5, p. 408 425] for additive functions of a set in R n . Among the many memoirs devoted to simplifying the proof we may mention: G. Paber fl], W. H. and G. C. Young [1], H. Stein- haus [1J, Ch. .1. de la Vallee Poussin fl, I; p. 103], A. Rajchman and S. Saks [1], J. Bidder [3] (cf. also the direct proof of Lebesgue's theorem for additive functions of bounded variation of an interval in the first edition of this book). More recently F. Riesz [6; 7] has given an elegant proof of Lebesgue's theorem for functions of a real variable. Finally S. Banach f4, p. 177] has exten- ded the theorem in question to a class of functions of an interval which is slightly wider than that of additive functions of bounded variation. The proof given here applies also without any essential modification to the theorem of Banach. For an extension of the theorem to certain abstract spaces, see J. A. Clark- son [1] (cf. also S. Bochner [3]) Another application of Lemma 5.1 is the proof of the fol- lowing theorem on term by term derivation of monotone sequences of additive functions: (5.7) Theorem. If an additive function of a set <P is the limit of a mono- tone sequence {$} of additive functions of a set, then almost everywhere = lim D$n(x. In the same way, if an additive function of bounded variation of an interval, F, is the limit of a monotone sequence {F n } of additive functions of bounded variation of an interval, then almost everyichere F'(x)=-limF', t (x). n Proof. Suppose, to fix the ideas, that the sequence {#} is non-decreasing and write & n =$ <P M . To establish the first part of the theorem, we need only show that (5.8) limD9,,(jp) = n almost everywhere. For this purpose, let A denote the set of the points x at which (5.8) is not satisfied and suppose that \A\ > 0. For any positive inte- ger k we write A* for the set of the points x at which lim Dfy,(#)^l/fr. n Since A~Ak, there is an index k=k Q such that |-4*J> 0. Let B de- ft note any hounded subset of A^ of positive measure and / an interval containing B. Since the sequence {9 n } is non-increasing, so is the sequence {D,,} and therefore we must have D(9, f (^r)^ l/fc for n = I, 2, ... at any point #e #C A** Hence by Lemma 5.1, we find [ 6] Derivation of the indefinite integral. 117 f r every positive integer n, and this is clearly a con- tradiction since |B| > and lim #(/) = 0(1) Urn $(!) = 0. n n We might prove similarly the second part of the theorem, but actually the latter can be reduced at once to the first part. In fact if we suppose the given sequence {F n } non-decreasing and write T n F F n , then the functions of an interval, T w , are non- negative and the sequence (T n (I)} converges to in a non-increasing manner for every interval I. The sequence {T*} of additive and non-negative functions of a set is then also non-increasing and con- verges to (cf. Chap. Ill, Theorem 6.2). Hence, by the first part of our theorem and by Theorem 2.1, we have lim7 T n(^)=limDTj}(-r)=0,, n n and therefore lim F' n (x)=F'(x), for almost all x, which completes n the proof. For functions of a real variable, Theorem 5.7 may be stated in the fol- lowing form (vide G. Fubini [2]; cf. also L. Tonelli [3] and F. Rienz [6; 7]): // F(x)=^F n (x) is a convergent series of monotone no n -deer easing functions, n then the relation F'(x)~^Fn(x) holds almost everywhere. $ 6. Derivation of the Indefinite integral. Given a set A, let us write L A (X) \A-X\ for every measurable set X. The function LA of a measurable set, thus defined, is termed measure-function for the set A. Considered as function of a measurable set, or as function of an interval, LA is additive and absolutely continuous; and, if further the set A is measurable, we have L A (X) = for every measurable set X, i. e. the function L A is the indefinite integral of the characteristic function of the set A. (6.1) Theorem. For any set A we have (6.2) at almost all x of A\ and if further, the set A is measurable, then (6.2) holds almost everywhere in the whole space. Proof. By Theorem 6.7 of Chap. Ill the set A can be enclosed in a set H e ^ for which the measure-function is the same as for the set A. Let us write JBr=/7fl ( n, where {O n } is a descending n CHAPTER IV. Derivation of additive functions. sequence of open sets. We clearly have DL# (#)=! at any point xeO n ^ and so a fortiori at any point xeACH(2G n . Hence, remembering that the sequence {L@ } of functions of a set is non-increasing and converges to the function L A =L Hj it follows by Theorem 5.7 that DL A (x) = lirn DL^ (x) = 1 = C A (X) almost everywhere in A. n n Now suppose the set A measurable. Then L A (X) + LcA(X)=\X\ for every measurable set J, and consequently DL A (x) + DLc A (x) = 1 at any point x at which the two derivatives DL A (x) and DLc A (x) exist, i. e. almost everywhere by Lebesgue's Theorem 5.4. Now, by what has been proved already, DLc^(^)==l almost everywhere in CA. Therefore DL A (x)-= Q==c A (x) at almost all a? of CA and this shows that (6.2) holds almost everywhere in the whole space. (6.3) Theorem. If (P is the indefinite integral of a summable junction /, then (6.4) at almost all points x of the space. Proof. We may clearly assume that / is a non-negative function. If / is the characteristic function of a measurable set, the relation (6.4) holds almost everywhere by the second part of Theorem 6.1. The rela- tion therefore remains valid when / is a finite simple function, i. e. the linear combination of a finite number of characteristic functions. Finally, in the general case, any non-negative summable function / is the limit of a non-decreasing sequence {f n } of finite simple measur- able functions; therefore, denoting by (D n the indefinite integral of /, it follows from Theorem 5.7 that D$(x) = lim D<P,,(o?) = Iimf 11 (x)^f(x) ii n almost everywhere, and this completes the proof. 7. The Lebesgue decomposition. In this we shall give for additive functions of a set ("#), a more precise form to the Lebesgue Decomposition Theorem 14.6, Chap. I. We shall prove in fact that the absolutely continuous function which occurs in this theorem is the indefinite integral of the general derivative of the given function. At the same time we shall establish the corresponding decomposition- for additive functions of bounded variation. [ 7] The Lebesgue decomposition. 119 (7.1) Lemma. If <P is a singular additive function of a set, then ) almost everywhere. Proof. We may assume (cf. Chap. I, Theorem 13.1 (1)) that the function is non-negative. The function $ being singular, there exists a set E Q measur- able (53) and of measure zero, such that (7.2) $(X-CE ) = for every set X bounded and measurable ( S B). Suppose that the set of the points x at which D$(#)>0 has positive measure. Then denoting by Q,, the set of the points xeCE at which !)$(#)> 1/n, there exists a positive integer jV such that KMX). Consequently, there also exists an interval I such that \I-Qv >0, and by Lemma 5.1 we find 0(1 C# )^ 0(1 * Q, which clearly contradicts (7.2). (7.3) Theorem. If is an additive function of a set, the derivative D0 is Hummable, and the function <P is expressible as the sum of its function of singularities and of the indefinite integral of its general derivative. Proof. By Theorem 14.6, Chap. I, we have $=:<9+2f where # is a singular additive function of a set and (F is the indefinite integral of a summable function /. Hence, making use of Theorem 6.3 and of Lemma 7 . 1 we find almost everywhere D $ (x) =D 8 (x ) + D W(x) =f(x) and this proves the theorem. We can extend the theorem to additive functions of bounded variation of an interval. We have in fact: (7.4) Theorem. If F is an additive function of bounded variation of an interval, the derivative F' is summable, and the function F is the sum of a singular additive function of an interval and of the in- definite integral of the derivative F'. Moreover , if the function F is non-negative, ive have for every interval I (7.5) /o equality holding only in the case in which the function F is absolutely continuous on I . 120 CHAPTER IV. Derivation of additive functions. Proof. We may clearly assume the function F to be non- negative in both parts of the theorem. The corresponding function of a set F*, together with its function of singularities, will then also be non-negative; and on account of Theorems 7.3 and 2.1 we shall have (7.0) P*( A') ^ fl>F*(x) dx = fF'(x) dx for every bounded set X e <B. x x. Let us write for any interval 1 (7.7) The function of an interval thus defined is clearly non-negative, since by (7.0) '(x) dx = F\x) dx for every interval 1. Moreover, if we take the derivative of both sidesof ( 7. 7) in accordance with Theorem 6. 3, we find DT*(x)~T'(x)=Q almost everywhere. It therefore follows from Theorem 7.3 that the function of a set T* and so by Theorem 12.6, Chap. Ill, the function of an interval T are singular. The relation (7.7) therefore provides the required decomposition for the function F. Finally, since the function T is non-negative, it follows from (7.7) that the inequality (7.5) holds for every interval / , and re- duces to an equality if, and only if, T(/) = for every interval /C^o- * n other words, in order that there be equality in (7.5) it is necessary arid sufficient that the function F be on / the inde- finite integral of its derivative, i. e. be absolutely continuous on / . Theorem 7.4 provides a decomposition of an additive function of bounded variation of an interval into two additive functions one of which is absolutely continuous and the other singular. Just as for functions of a set, this decomposition is termed Lcbengue decomposition and is uniquely determined for any additive function of bounded variation. For suppose that Grf T^G^T^ where G l and G 2 are absolutely continuous functions and T l and T 2 are sin- gular functions; then G 2 0^=T l T 2 and by Theorem 12.1 (2, 6), Chap. Ill, this requires G l = G 2 and T l =T 2 . The absolutely con- tinuous function and the singular function occurring in the Lebesgue decomposition of a function of bounded variation F are called, respectively, the absolutely continuous part and the junction of sin- gularities of the function F. [ 8] Rectifiable curves. 121 As a special case of Theorems 7.3 and 7.4, let us mention the following result, in which part 2 includes Lemma 7.1 and its converse. (7.8) Theorem. 1 An additive junction of a set, or an additive function of an interval of bounded variation, is absolutely continuous, //, and only i/, it is the indefinite integral of its derivative. 2 An additive function of a set, or an additive function of an interval of bounded variation, is singular if, and only if, its derivative vanishes almost everywhere. Finally, let us mention also an almost immediate consequence of Theorems 7.3 and 7.4: (7.9) Theorem. The derivative of the absolute variation of fin ad- ditive function of a set, or of an additive function of an interval of bounded variation, is almost everywhere equal to the absolute value of the derivative of the given function. Proof. Consider to fix the ideas, an additive function of bound- ed variation of an interval, F. Let T be the function of singular- ities of F, and let W and V be the absolute variations of the func- tions F and T respectively. In virtue of Theorem 7.1 the relation F(I) ~ I F'(x)djc + T(I) holds for every interval Z, and hence also (7.10) W(I)^f\F'(x)\dx Now the function V is singular together with T, so that its deriva- tive vanishes almost everywhere by Lemma 7.1. Hence taking the derivative of (7.10), we find on account of Theorem 6.3 that W'(x)^\F'(x)\ almost everywhere, and this completes the proof since the opposite inequality is obvious. 8. Rectifiable curves. By a curve in a space R m we shall mean any system C of m equations x,=X,(t) where i=l,2, ... , w and the Xt(t) are arbitrary finite functions defined on a linear interval or on the whole straight line JK r The variable t will be termed para- meter of the curve. The point ( J^t), A' 2 (J), ... , X m (t)) will be called point of the curve corresponding to the value t of the parameter, and denoted by p(C; t). If E is a set in IZ^ the set of the points p((7; t) for teE will be called graph of the curve C on E and de- noted by B(C; E) (cf. the similar notation for graphs of functions, Chap, in, p. 88). 122 CHAPTER IV. Derivation of additive functions. For simplicity of wording we consider in the rest of this , only curves in the plane R 2 \ we shall suppose also that the functions determining these curves are defined in the whole straight line B v But needless to say, these restrictions are not essential for the validity of the proofs that follow. Let therefore C be a curve in the plane, defined by the equa- tions x=X (t), y Y(t). Given any two points a and ft>a, a finite sequence T = {fy}j = o,i,...,n of points such that a = tQ^t l ^...^t n =b will be called chain between the points a and ft, and the number 29(pj-i,Pj), where Py=p(<7; ty), y=i will be denoted by o(C;t). (We may regard this number as the length of the polygon inscribed in the curve C and whose vertices correspond to the values a=t oj t^ ... , t n = b of the parameter.) The upper bound of the numbers 0(C;i) when i is any chain between two fixed points a and ft, will be called length of arc of the curve C on the interval I [a, ft], and will be denoted by S(C; I) or S(C; a, ft). If S(<7; I) ^ oo the curve C is said to be rectifiable on the interval /; and if this is the case on every interval we say sim- ply that the curve C is rectifiable. (8.1) For any curve C we have 8(C; a, ft)+S(C; ft, c) = S(C; a, c) whenever a<ft<c. It is enough to prove that S(C; a, ft) + S(O, ft, c)^S(C; a, c), since the opposite inequality is obvious. Let T={a=t Q , t v ... ,/7=c) be any chain between a and c, and let h be the index for which th~\<,b<t h . Writing T 1 ={a=* ,t ll ...,fe-i,6} and ^={b j t hj ... j t n -=c} 1 we have a(C; T)<a(C; r 1 ) + a(0; T 2 XS(C; a, ft)-f S(C; ft, c), and so S(C; a, c)<S(C; a, ft) + 8(0; ft, c). It follows from (8.1) that if a curve C is rectifiable, the length of arc S((7;/) is an additive function of the interval /. We shall call this function length of the curve C. Any function of a real vari- able that corresponds to this function of an interval, i. e. any function 8(t) such that S(b) #(a)=S(C; a, ft) for every interval [a, ft], will also be termed length of the curve C. [8] Rectifiable curves. 123 (8.2) Jordan's Theorem. If C is a curve given by the equations x^X(t), y=Y(t), we have C; I), (8.3) for any interval I; and therefore, in order that the curve C be rectifiable on an interval I , it is necessary and sufficient that the functions X and Y be of bounded variation on Z . Proof. Given an interval /== [a, 6], we easily find that for any chain ?= {at oj t 19 ... , t n b] between the points a and fc, jj \X(tj) - y-i y-i from which the inequalities (8.3) follow at once. (8.4) Theorem. If C is a rectifiable curve given by the equations x~X(t), y= (t)j and the function S(t) is its length, then (i) in order that S(t) be continuous at a point [absolutely contin- uous on an interval] it is necessary and sufficient that the two func- tions X(t) and Y(t) should both be so; (ii) we have A(B(C; E)}^\8[E]\ for any linear set E (i.e. on any set E the length of the graph of the curve C does not exceed the measure of the set of values taken by the function S(t) for teE), (iii) [S'(t)] 2 =[X'(t)]* + [J'(t)] 2 for almost every J; b (iv) S(C-,a,b)^fVW'(t)]^[YWjdt for every interval [a, b], a and the sign of equality holds if, and only if, both the functions X(t) and Y(t) are absolutely continuous on [a, b]. Proof, re (i): This part of the theorem is an immediate con- sequence of the relations (8.3), since by Theorems 4.8 and 12.1 (1) of Chap. Ill, a function of bounded variation is continuous at a point [absolutely continuous on an interval] if, and only if, its absolute variation is so. 124 CHAPTER IV. Derivation of additive functions. re(ii): Let e be any positive number and let {I n } be a sequence of intervals such that (8.5) 8[E]Cl and (8.6) |7,,| " n We may clearly assume that no interval I n has a diameter exceeding *. This being so, we write ^=B(T; E) and we denote by E n the set of the points p(C;t) of the curve C for each of which S(t)eln. It is easy to see, on account of (8.5), that EC^En. On the other hand d(En)^A (l rl )-=\I n \^E for any w, and it follows from (8.6) (cf. Chap. II, 8) that J, ( <H)<|tf[^J| + *, and hence by making e tend to zero, that A(E) ^ |#[#]|, as asserted. ?'f(iii): Let / = [ ? *ol ^ e an Y interval. We shall show suc- cessively that both the relations (8.7) [S'(t)]*^[X f (t)Y+[Y'(t)]* and (8.8) hold almost everywhere in / ; here the derivatives 8'(t), X'(t) and Y'(t) exist almost everywhere on account of Theorem 8.2 and of Lebesgue's Theorem 5.4. We have 8(t + h)-S(t)^{[X(t + h)-X(t)? + [Y(t + h)-r^^ for any point t and any h > 0, and if we divide both sides by h and make h -> 0, this implies the relation (8.7) at any point t for which all three functions are derivable at the same time, i. e. almost everywhere. NTow let A denote the set of the points t f / at which the three derivatives X'(t), Y'(t) and S'(t) exist without satisfying the inequal- ity (8.8); and for any positive integer w, let A n denote the set of the points t e A for each of which the inequality 8(I)I\I\ holds for all the intervals I containing t, whose diameters are less than 1/n. Clearly A=A n . n Keeping n fixed for the moment, let e be any positive number. There exists a chain T= {a = t , t^ ... , t p = b } such that t k fo-i <l/n for /c=l, 2, ... , p, and such that S(C;I )^ a (C f ;' r )+ f - Consequently, writing for brevity </*=[<*__ <J, P A =P(C;< A ), ^= e(p k _ l9 p k ), we have (8.9) S(Jk)l2*Gk+\Jk\ln for fr=l,2,... ,p, whenever and on the other hand (8.10) [ 9] De la Valtee Poussin's theorem. 125 Therefore if (n) stands for a summation over all the indices k for which Jk>A n ^ 0, we find on account of (8.9) and (8.10) that -^ Now since e is an arbitrary positive number, it follows that [JL^I^o for any n, and therefore also that |^4| 0. Thus the in- equality (8.8) holds almost everywhere, as well as the inequality (8.7), and this completes the proof of part (iii) of the theorem. Finally, since S(C; a, b)=S(b) S(a), part (iv) reduces on account of (i) and (iii) to an immediate consequence of Theorem 7.4. Theorem 8.4 (in particular its parts (iii) and (iv)) is due to L. Ton ell i [1; 4]; cf. also F. Riesz [6; 7]. As regards part (ii) of the theorem, it may be observed that in the ease in whieh the curve C has no multiple points (i. e. when every point of the curve corresponds to a single value of the parameter) the inequality A[E(C; E)} \8[E]\ can easily be shown to reduce to an equality. As proved by T. Wazewski [Ij, any bounded continuum Pot finite length may be regarded as the set of the points of a curve C on the in- terval [0, 1], such that S(C;0,1) 2- - 9. De la Vall6e Poussln's theorem. With the help of the results established in the preceding we can complete further, for continuous functions of bounded variation of a real variable, the decomposition formula of Lebesgue. We shall begin with the following theorem, which itself com- pletes, in part at any rate, the second half of Lebesgue's Theorem 5.4. (9.1) Theorem. If F(x) is a function of bounded variation and W(JT) denotes its absolute variation, then for the set N of the points at which the function F(x) is continuous but has no derivative finite or infinite, we have (9.2) |F*(tf)|=lf%yH |*1 = and (9.3) A(K(F; A')) = 0. Proof. Consider the curve C: x=x, y=F(x). Let S(x) be the length of this curve and let E be the set of the values assumed by the function 8 (#). For any se E, denote by A'(ir) the value of x for which 8(x) = s and write Y(s)=F(X(s)). Since (cf. Theorem 8,2) |-T(*,) -r(i)|<| t *j| and |T> 2 ) Y^Kta *J for any pair of points lt * t of E, the functions -V(*) and Y(*) are continuous 126 CHAPTER IV. Derivation of additive functions. on the set E, and moreover may be continued on to the closure E of this set by continuity. If now [a, b] is an interval contiguous to E, the function X(s) assumes equal values at the ends of this interval, and the point xX(a)~X(b) is a point of discontinuity of the function F(x). We shall complete further the definition of the functions X(s) and Y(s) on the whole straight line J^ so as to make the former constant and the second linear, on each interval contiguous to E. This being so, consider the curve C l given by the equations x=X(s), y = Y(s). We verify easily that the parameter 8 of this curve is its length. (Actually we see easily that the graph of the curve C l is derived from that of the curve C by adding to the latter at most an enumerable infinity of segments situated on the lines x=d where d are the points of discontinuity of the function F.) By Theorem 8.4 (iii), we therefore have [X f (s)] l + [Y'(s)] 2 =l for almost all s, and therefore the set H of the points 8 for which either one of the derivatives X'(s) and Y'(s) does not exist, or both exist and vanish, is of measure zero. Now we see at once that if s e E ff, then at the point #= X(s), the derivative F'(x) exists (with the value Y'(s)IX'(s) # JT(*)4=0, or with the value 00 if X'(s) = and Y'(s)^O). Therefore N(2X[E'H], or what amounts to the same, S[N](2E-H, an< i hence S[NJ^\H\ = 0. From this we derive at once with the help of Theorem 8,4 (ii) the relation (9.3), since B(jP; jy r )=B(C; N). Finally, the function 8(x) is continuous (cf. Theorem 8.4 (i)) at any point at which the function F(x) is continuous, and so at any point of the set N, and therefore it follows from Theorem 13.3, Chap. Ill, and from Theo- rem 8.2, that |F*(-y)KW*(^XS*(jy)=|S[^]| = 0; this completes the proof. (9.4) Lemma. If F(x) is a function of bounded variation, then (i) F*(A)^k-\A\ for any bounded set A and any finite number k whenever the inequality F'(x)^k holds at every point x of A (<md the assertion obtained by changing the direction of both inequalities is then evidently also true), (ii) F*(B)=sQ for any bounded set B of measure zero throughout which the derivative F'(x) exists and is finite. [I 9] De la Vallee Pouwin'a theorem. 127 Proof, re (i). Let e be any positive number. Denote, for any positive integer n, by A (n) the set of the pofnts xeA such that F(I)^(k e).|J| holds whenever I is an interval containing x and of diameter less than 1/n. Clearly A = lim ^4 (n) . n Keeping the index n fixed for the moment, let us denote by (n} a bounded open set containing A (n \ such that (cf. Theorem 6.9, Chap. Ill) (9.5) \F*(X) F*(A (n} )\<e whenever A (n) C^CG (n \ and let us represent (n} as the sum of a sequence {lj> n) }/>=i,2,... of non- overlapping intervals. We may clearly suppose that all the inter- vals I ( p } are of diameter less than 1/n and that their extremities are not points of discontinuity of F(x). So that if ^ stands for P summation over the indices p for which 7J, n) A (n} =f= 0, and X 00 denotes the sum of the intervals I ( ^ corresponding to these indices, we find J"(Z<">)=2?"!F(^ and p P hence on account of (9.5), F*(A (n) )^(k t)-\A (n} \ e. Making and n-oo, we obtain in the limit F*(A)^k- l A\. re (ii). Let JB (n) denote the set of the points x B for which F'(x)^ n. By(i)wehaveP*(B (ri) )^ n-| (n) | = for any positive integer n, and so J?*(J5)>0. By symmetry we must have also JF*(jB)<0, and therefore jF*(B)=0. (9.6) X>e la Valtee Poussin's Decomposition Theorem. If F(x) is a function of bounded variation with W(x) for its absolute varia- tion, and if E+& and _> denote the sets in which F(x) has a deriv- ative equal to +00 and to oo respectively, then (i) for any bounded set X measurabh () at each point of which the function F(x) is continuous, we have the relations: (9.7) (9.8) W*(X)=F*(X-E +X> ) + \F*(X-E^)\ + f\F'(x)\dx; (ii) the two derivatives F'(x) and W'(x) exist and fulfil the rela- tion W(x)=\F'(x)\ at any point x of continuity of F, except at most at the points of a set N such that W*(N)=\N\ = Q (i.e. a set which is at the same time of measure (L) zero and of measure (W) zero). 128 CHAPTER IV, Derivation of additive functions. Proof. re(i). On account of Theorem 7.3 there exists a set A of measure zero such that for any set A' bounded and measurable (9.9) A' Supposing further that the function F is continuous at every point of A', we may assume, in virtue of Theorem 9.1, that the function F(x) has everywhere in A, a derivative, finite or infinite. Moreover since by Lemma 9.4 (ii) the function F*(X) vanishes for any bounded subset of A (E+OQ+ E~oo), we may assume that A(^_E 1.^+E-^. Finally, we obtain directly from (9.9) that F*(X) vanishes when X(^CA and |A'| = 0. We may therefore choose simply A =E +*,>-}- E < and formula (9.9) becomes (9.7). Formula (9.8) is, by Theorem 13.2, Chap. Ill, an immediate consequence of (9.7), since Lemma 9. 4 (i) shows that and F*(X-E oo)^0 for every set A' bounded and measurable re(ii). Let N be the set of the points *r at which the function F(x) is continuous and either one at least of the derivatives F'(.r) and W(x) does not exist, or both exist but do not satisfy the re- lation W'(jr)=\F'(jr)\. We then have A r -i?+oo = N-E- 00 = 0, since evidently W'(x)=+oo=\F'(x)\ at any point where F'(x) = oo. Therefore, since the set X is further, by Theorem 7.9, of measure (L) zero, it follows from formula (9.8) that W*(j\ r ) = Q and this com- pletes the proof. Let us mention an immediate consequence of Theorem 9.6. In order thai a continuous function of bounded variation F(JC] be absolutely continuous, it IK necen- Qary and nufficie-nt that the function of a set F*(X) should rani'trA identically on the set of the point** at which F(JT) hats an infinite derivative. In particular therefore. any continuous function of bounded variation which is not absolutely continuous has an infinite derivative on a non -enumerable set. Let us remark further that the theorems of this cannot be extended directly to additive functions of an interval in the plane. Thus if F(l) denotes the con- tinuous singular function of an interval in /f 2 , which for any interval / equals the length of the segment of the line x y contained in /, we have F(JT) -0 for every point jr. so that F'(jr)=oo does not hold at any point. $ 10* Points of density for a set. Given a set E in a space lt fn , the strong upper and lower derivates of the measure-function of E (cf. 6, p. 117) at a point x will be called respectively the outer upper and outer lower density of E at x. The points at which these two den- sities are equal to 1 are termed points of outer density, and the points at which they are equal to 0, points of dispersion, for the set E. L 10] Points of density for a set. 129 If the set E is measurable we suppress the word "outer" in these expressions. We see further, that if the set E is measurable, any point of density for E is a point of dispersion for CU, and vice- versa. We shall show in this (cf. below Theorem 10.2) that almost all points of any set E are points of outer density for E, or what amounts to the same that (10.1) For almost all points x of E, if {!} is any sequence of intervals tending to x (in the sense of 2, p. 106), we have \E-I n \/\I n \-+l. This proposition presents an obvious analogy to Theorem 6.1 and it is in the form (6.1) that the "density theorem" is often stated and proved either with the help of Vitali*s Covering Theorem or by more or less equivalent means (vide, for instance, E. W. Hobson [I], Ch. J. de la Vallee Poussiu [I, p. 71J and W. Sierpinski [101). Theorem 10.2 will be however, so to speak, independent of Vitali's theorem, because the sequences of intervals occurring in (10.1) are not supposed regular. Also, Theorem 10.2 will be more precise than Theorem 6.1, because at any point x of outer density for a set E we have a fortiori DL K (x)=l, and in- deed, at any x the relations L A ,(#)-1 and D}j K (x)l are equivalent. To see this it is enough to show that the former of these relations implies the latter, the converse being obvious from Theorem 2.1. We may assume moreover, on account of Theorem 6.7, Chap. Ill, that the set E is measurable. Let therefore x .be a point such that L K (x )^l and let (X n \ be a regular sequence of measurable sets tending to # . Then there exists a sequence {J n } of cubes such that X n J n and \X n \j\J n \>a, where w~^l,2, ... and a is a fixed positive number. Since by hypothesis, \E'J n \l\J n \ >l, and HO \CE*J n \l\J n \*Q f it follows that \CE-X n \l\X n \->0, or what amounts to the same that \E- X n \l\X n \-+l. Therefore DLg(a; ) = l. It is, of course, only for spaces H m of dimension number m ^ 2 that Theo- rem 10.2 will differ from Theorem 6.1. The two statements are equivalent for K lm For the various proofs of Theorem 10.2 vide F. Riesz [8J and H. B use- in a nn and W. Feller [1]. In the second of these memoirs will be found a general discussion of the different forms of "density theorems". It is of interest to observe that the proposition (10.1) ceases to be true even for closed sets E, if the intervals I n are replaced by arbitrary rectangles with sides not necessarily parallel to the axes of coordinates. This remarkable fact has been established by O. Nikodyrn and A. Zygrnund (vide O. Nikodym [1, p. 167]) and by H. Busemann and W. Feller [1, p. 243]. (10.2) Density Theorem. Almost all the points of an arbitrary set E are points of outer density for ; and if further the set E is measurable, almost all the points of CE are points of dispersion for E. Proof. For simplicity of notation, we shall state the proof for sets which lie in the plane; the corresponding discussion in any space Ji m is however essentially the same (except that in lt^ as already remarked, the theorem reduces to Theorem 6.1). 130 CHAPTER IV. Derivation of additive functions. By Theorem 6.7, Chap. Ill, any set can be enclosed in a meas- urable set having the same measure-function. On the other hand any measurable set is the sum of a set of zero measure and of a sequence of bounded closed sets. We may therefore suppose that the set considered is bounded and closed. Let e be any positive number. We shall begin by defining a positive number a and a closed subset A of E such that \E and that, for any point (, *?) in the plane, (1 ) (b a) (i) whenever (, y)A, a^^b and b a To do this, let us write for brevity, when Q is any set in the plane, $ [y/] =E[(#, y) eQ], and let us denote, for any positive X integer n, by A n the set of the points (, y) of E such that |jB l ^-Z|^(l e)-\I\ whenever I is a linear interval containing f and of diameter less than 1/n. The sequence {A n } is evidently ascending. Let us write (10.3) N=E Urn A n . n For any 17, if is a point of the set N ] , there will then exist, for each positive integer n, a linear interval I such that e/, d(I)<l/n and l^-JTKIJB^-I^d c)-\I\. Therefore the lower derivate of the measure-function for the linear set N [)I} cannot exceed 1 e at any point of this set, whence by Theorem 6.1, (10.4) JJV ||?1 |=|E[(0, r})eN]\Q for every real number r\. X Let us now remark that the sets A n are closed. For, keeping for a moment an index n fixed, let ( , *7 ) be the limit of a sequence {(/ r 7 t ? A )}/ f= , 1>2 , f points of A n . Let J be a linear interval such that 0*^0 an( i ( '(J r o)< J / n ? an( i I 6 *- 1 be any linear interval con- taining I in its interior, whose diameter is less than 1/n. Then for any sufficiently large fc, ^el and therefore | C ^ ] ./|Xl )-|/|. On the other hand, the set E being closed, we easily see that J^^^limsupJS 1 ''* 1 ; so that \E [ ^-1 ^ lim sup | [/?/fl -11^(1 e)-|/|, A * and therefore also 1 E M -I Q '^(1 ^)-|I - Hence (^ , ?; )6 A n , i.e. each set A n is closed. It follows, according to (10.3), that the set N is measurable. [ 10J Points of density for a set. 131 Therefore, applying Fubini's theorem in the form (8.6), Chap. Ill, we conclude, on account of (10.4), that the plane set N is of meas- ure zero. Consequently \E J./J < e for a sufficiently large index n , and writing o=l/n and A=A^ we find that the inequality \E A\< and condition (i) are both satisfied. In exactly the same way, but replacing the set E by A and interchanging the role of the coordinates x and y, we determine now a positive number a l < o and a closed subset B of A such that \A J?|<c and that, for any point (, 17) in the plane (i *)( a) (u) 9 whenever (, *?)6.B, a^f?^ft and b a This being so, let (f , %) be any point of B. Let t7=[i, ft; 2 j ft] denote any interval such that (^^)eJ and <J(t7)<<7 1 <a. By Fubini's theorem (in the form (8.6), Chap. Ill) we have /* (10.5) E.J\=J\E[(x,y)cEi Since ( , /y )e.B and c^^^o^ft* ^ follows from (ii) that the set of the y such that ( , y)eA and 2 ^y^ft is of measure at least equal to (1 f) (ft a 2 ). On the other hand, since i<^ ^ft> it follows from (i) that |E[(#, y) e E\ !<a?<ft] >(1 )(ft a x ) whenever ( , y)eJ.. Hence, formula (10.5) gives i. e. the lower density of jB is at least equal to (l-~) 2 at any point (<%) of B - Therefore, since \E B <|JS A +\A B\^2e and since e is an arbitrary positive number, the lower density of E is exactly equal to 1 at almost all the points of the set E, i. e. almost all points of this set are points of density for it. The second part of the theorem is an immediate consequence of the first part. In fact, if the set E is measurable, so is the set C?, and almost all points of CE, since they are points of density for CJ5, are points of dispersion for E. In connection with the definition of points of density, Denjoy introduced the important notion of approximate continuity of a function. We call a function of a point f(x) (in any space JK m ) 9 132 CHAPTER IV. Derivation of additive functions. approximately continuous at a point # , if f(x )=^oo and as x tends to # on a measurable set E for which # is a point of density. (10.6) Theorem. If f is a measurable function almost everywhere finite on a set E, then the function f is approximately continuous at almost all points of E. Proof. On account of Lusin's theorem (Chap. Ill, 7), given any > 0, we can represent the set E as the sum of a closed set F on which the function / is continuous and of a set of measure less than *. The function / is clearly approximately continuous at any point of density for the set JP, and so, by Theorem 10.2, at almost all points of F. This implies, as e is an arbitrary positive number, that the function / is aproximately continuous at almost all points of the set E itself. Theorem 10.6 is due to A. Denjoy [5] (cf. also W. Sierpinski [6; 91). It is easy to see that the converse holds also, i. e. that every function which is approximately continuous* at almost all the points of a measurable set E is measurable on E (vide VV. Stcpsinoff [2] and E. Kamke [1]). Let us mention also the following theorem, an almost im- mediate consequence of Theorem 10.6, which completes, in part, Theorem 2.2 on derivation of an indefinite integral: (10.7) Theorem. If is the indefinite integral of a bounded meas- urable function /, then $((x)~f() at almost all points x, and in fact, at any point x at which the function f is approximately continuous. Proof. Let x be a point at which the function / is approx- imately continuous and let E be a measurable set for which X Q is a point of density, while /(#)-> f(x ) as x tends to X Q on the set E. We may suppose (by subtracting, if necessary, a constant from /(a?)) that /(. )=0. Therefore, given any positive number f, we have for any interval / of sufficiently small diameter con- taining x , (i) I'CE^i'I and (ii) f(x) <E for every xel-E. Denoting by M the upper bound of |/(a?)|, conditions (i) and (ii) imply |<P(/)| < \9(I.QE)\+9(I-E)\ < Me- |Z|+f-|I| < e-(M+l)-\I\, whence <^(# )==0=/(# ). If # is the indefinite integral of a function / which is summable but un- bounded, it may happen that the relation $<(x)^=f(x) is not fulfilled at any point. In virtue of Theorem 6.3, this relation clearly holds at almost all points at [ 11] Ward's theorems. 133 which the strong derivative $s(x) exists; but if the function / is unbounded, its indefinite integral <fl may have no finite strong derivative at any point (cf. on this point Busemann and Feller fl, p. 256]; the result of Banach and Bohr mentioned above, p. 112, follows as a particular case). Nevertheless, the result contained in Theorem 10.7 may be generalized considerably. In fact, according to a theorem of B. Jessen, J. Marcinkie- wicz and A. Zygmund [1] (see below 13) the indefinite integral of a function f in a space R m is almost everywhere derivable in the strong sense whenever the function l/Klog*!/!)" 1 "" 1 w summable (in a less general form, for functions / of which the power p > 1 is summable, this theorem was established a little earlier by A. Zygmund [1]). On the other hand however, given an arbitrary function o(t) positive for t>0 and such that lim inf o(t}~ 0, there always exists a junction f(x) /-H-oo in R m such that the junction ^d/IH/Klog"*"!/!) 111 ~ ! w summable and such that the indefinite integral of f is not derivable in the strong sense (has the strong upper derivate +00) at any point of R m * * 11. Ward's theorems on derivation of additive func- tions of an interval. In the preceding of this Chapter, we have treated the Lebesgue theory of derivation of additive func- tions of an interval of bounded variation. As regards functions of a real variable, this theory has been extended to arbitrary func- tions by Montel, Lusin and especially by Denjoy. Recently Denjoy 's theorems, which already belong to the classical results of the theory, have been generalized still further. On the one hand they have been given a geometrical form by which they become theorems on certain metrical properties of sets, and in this form an account will be given of them in Chapter IX. On the other hand, recent researches of Besicovitch and Ward have made it possible to extend an essential part of the Denjoy results, particularly the relations between the extreme bilateral derivates, to additive functions of an interval in a space R m of any number of dimensions. These researches will form the subject of the present . It was A. S. Besicovitch [5] who started these researches, by establishing between the extreme strong and ordinary derivates of absolutely continuous functions of an interval, relations analogous to those proved by Denjoy for derivates of functions of a single variable. A. J. Ward [2; 5] has extended this result to quite arbitrary additive functions of an interval. Of the two theorems of Ward (vide, below, Theorems 11.15 and 11.21) one concerns only ordinary derivates, while the other applies also to strong derivates. It is the latter that generalizes the result of Besicovitch; this second theorem is one which can be proved fairly simply for functions of an interval in the plane; it is rather curious that it requires much more delicate methods in an arbitrary space lt m . 134 CHAPTER IV. Derivation of additive functions. We shall make use in this of some auxiliary notations. If F is an additive function of an interval and a ^ 1 is a positive number, F ( a)(x) and F()(x) will denote at any point x the upper and lower limit of the ratio F(I)/\I\ where Z is any interval containing #, which is subject to the condition r(/)^a, and which has diameter tending to zero. We see at once that at any point a?, F( n) (x) and F (a )(x) tend to F(x) and F(x) respectively as a->0. We shall suppose fixed a Euclidean space R m , and in it we define a regular sequence of nets of cubes {Q*}*=--i,2 f ..., denoting by {Q/r! the family of all the cubes of the form [P,2~ A ,(P 1 + 1)2-*; p 2 2-*,(p 2 +l)2-*;...; p m 2~*, (p m +l)2~ k ] where p,, p 2 , ..., p m are arbitrary integers. (11.1) Jemima. Given an additive function of an interval (?, and positive numbers a ^2~ m and a, suppose that the inequalities 0<G ( )(x)<a hold at every point x of a set E having positive outer measure; then there exists for each e > 0, a cube Q which belongs to one of the nets Q* and for which we have (11.2) <J(Q)<e, \E.Q\X1 E).\Q\ and G(Q)<8 m -a~ m >a.\Q\. Proof. By replacing, if necessary, the set E by a suitable subset of E having positive outer measure, we may suppose that there exists a positive number o such that for every interval Z, (11.3) (?(/)> whenever I-E^Q, r(Z)^a and 6(I)<a. We may further clearly assume tnat a is less than both e and a m /8 m . This being so, let x^eE be a point of outer density for the set E (cf. 10). Since Q (a \(x ) < a, we can determine an interval J=[a l , ft,; ...; a m , b m ] containing X Q and such that (11.4) (5(c/)<o, r(J)^a, \E.J\>(1 o*)-\J\ and G(J)<a-\J\. It follows in particular that (11.5) \E-I\>(1 a)-\I\ for any interval ZCV suchthat \I\>o-\J\. Let I be the smallest of the edges of J. Since r(J)^a, no edge of J can exceed I/a, and therefore we have J ^l m l<* m . Finally let k be the positive integer given by [11] Ward's theorems. 135 (11.6) l/2*<Z/4<l/2*- J , and let <?=[ai,6i; ...}a m9 b' m ] be a cube which belongs to the net Q* and which contains the centre of the interval J. By (11.6) we find that QC J and that a t a/^ J/4, fe, fc',^/4 and ft'/ a',=l/2*^/8. It follows easily that the figure J0Q can be subdivided into a finite number of non-overlapping intervals with no edge smaller than Z/8. Now any such interval can clearly be further subdivided into a finite number of non-overlapping subintervals whose edges all lie between 1/8 and Z/4. We thus obtain a subdivision of the figure JQQ into a finite number of non-overlapping intervals, whose para- meters of regularity are greater than, or equal to, 2~ m ^a, and whose volumes are greater than, or equal to, 8~ m r^8~ m a m .|t7|>cr-|J|. It therefore follows from (11.5) and (11.3) that (11.7) 0(JQ Q)>0. Similarly, it follows from (11.6) that |#|=2~* I71 ^8~T> ^8~ m a 171 . | J|>a-| J|, whence by (11.5) we derive at once the second of the relations (11.2); at the same time, by the relations (11.7) and (11.4), Q(Q)<G(J)<a-\J\^8?*~ m a-\Q\, and this gives the third of the relations (11.2) and completes the proof. (11.8) Lemma.. Lei G be an additive function of an interval in K mj E a set in S mj Q a cube belonging to one of the nets Q*, a/nd a > 0, e > and b arbitrary fixed numbers. Suppose that (i) |J0.0|>(1 ).|0| f (ii) G(I) > for every interval I such that I(ZQ, I-E^Q and _ (iii) G( a )(x)>b at any point xeE; then G(Q)>I2-~ m -a m b.(I 2 m t)-\Q\. Proof. We may clearly assume that the set E is contained in the interior of Q and that every point of the set is a point of density. This being so, we shall begin by establishing the following result: (11.9) Given any 17 >0, we can associate with any point xeE ' cube Pj belonging to one of the nets Q*, and a cube JDP, such V (a) <7(P)>a m .12- m .ft.|* > l and (b) ueJ, d(J)<ri and |J|=3 m .|P|. 136 CHAPTER IV. Derivation of additive functions. For this purpose, let us associate with the point x an interval 8 such that xeS, <J(S)<l/4, r(8)^a and G(S)>b-\S\. Let h denote the largest edge of $, and let k v be the positive integer satisfying the inequality 1/2* 1 > 2A>1/2 M ~ 1 . Let ^ be a cube of the net Q A| having points in common with $, and let J denote the cube formed by the 3 m cubes of the same net (including the cube 8 l itself) which have points in common with S v The cube J clearly contains the interval $, and since no edge of 8 can be less than a A, we find that (11.10) |J|-3^^2" m1 -6 m .2~ On the other hand, since 2 *' fe>2~ ( * 1+1) and aA>a-2 ( * !+2) , the figure JQ8 can be subdivided into a finite number of non-over- lapping intervals with edges greater than, or equal to, a-2 <A|4 2) , and therefore, as in the proof of Lemma 11.1, into a finite number of non-overlapping intervals whose edges have lengths between a -2 ( * l+2) and a- 2 ( *'~ M) . Therefore, denoting by I any interval of this sub- division, we find r(I)>2~ m and \I\^a m --2~ m(k > +2} = I2' m a m *\J\. Consequently, by supposing the interval . S, and a fortiori the cube J, sufficiently small, we may assume that d( J) < r\ and that each of the intervals of the subdivision in question contains points of E. It follows, by condition (ii) of our lemma, that G(JeS)>0, and so, by (11.10), that G(J)>Q(S)>b-\8\^12 - m -a m -b-\J\. Thus among the 3 m cubes of the net Q*, which make up the cube </, there is one at least, P say, such that G(P) > 12 ~ m .a-b-\P\, and the cubes P and /DP, thus defined, clearly satisfy the con- ditions (a) and (b) of (11.9). It now follows, on account of (11.9) and condition (i) of the lemma, that (with the help of Vitali's theorem in the form (3.8)) we can determine in Q a finite system of non-overlapping intervals P 19 P 2 , ... , P n belonging to the nets Q/,, such that: (11.11) O(P t )> 12~ m .a m &.UP,| and p r E^Q for t=l,2,... n, (11-12) I\P<\>(1 0-WI- Among the cubes of the nets Q*, we shall consider specially two classes of cubes. A cube of a net Q* contained in Q will be said to be of the first class if it is one of the cubes P^Pi, ..., P A ; and of the second classifit contains points of E and if further [ 11] Ward's theorems. 137 among the 2 m cubes (Q*+i) composing it, there exists at least one which does not overlap with any cube P/. Since the number of cubes P/ is finite, there exists a net QK such that no cube of this net contains cubes of the first class. Let 21 be the set of all the cubes of the first or second class contained in Q and belonging to the nets Q/t for k^K. The set of these cubes covers the whole cube Q. For if not, there would certainly exist in the net QA- a cube I C<? n t con " tained in any cube (21). Now, since 7 contains no cube of the first class, I would not contain ^ny point of the set JB; and since, by hypothesis, I is not contained in any cube of the first or second class, we could, starting with / , form in Q a finite as- cending sequence of cubes without points in common with E and which belong respectively to the nets Q#, Q/r-i, ... , Q^, where Q^ is the net containing the cube Q. But the last term of this sequence of cubes is evidently the cube Q itself, and we arrive at a contra- diction since E(^Q. Let us now remark that since all the cubes (21) belong to the nets of the regular sequence {Q*}, it follows that, of any two over- lapping cubes (21), one is always contained in the other. Hence, we can replace the system of cubes 21 by another system 2l 1 C < 2t which also covers <?, and which, this time, consists of non-over- lapping cubes. Let A be the sum of the cubes (21J of the first class. On account of (11.11) we have (11.13) 0(A) ^ 12~ m .a m .&.|^l- Moreover, since the figure QQA is formed of a finite number of cubes of the second class which do not overlap, it follows from condition (ii) of the lemma that (11.14) 0(QQA)^0. Finally, in each cube I of the second class, there is always n a cube whjch is contained in QQ^Pt and whose volume is 2~ m -|/|. il It therefore follows from (11.12) that \QQA\<2 m -f-\Q\, and in virtue of (11.14) and (11.13) we find which completes the proof. (11.15) Theorem. Any additive function of an interval F_is derivable at almost all the points x at which either F(x) > oo, or F(x) < + oo. 138 CHAPTER IV. Derivation of additive functions. Proof. Consider the set of the points x at which F(x)> oo and suppose, if possible, that the set A of the points x at which F(x)>F(x)> oo is of positive measure. We could then determine a number a>0, and a set BA of positive outer measure, such that F(a)(x) ^ oo and F( a )(x) F(<*)(x) > a at every point x of B. We may clearly assume that a^2~ m . Let e be any positive number. Let us denote for any integer p- by B p the set of the points x of #at which pe <^F( a )(x)^(p + l)ej and let PO be an integer such that \B pit \ > 0. We can determine a number a>0 and a set E(^B P ^ whose measure is not zero, so as to have F(I) > p e- 1/| whenever the interval / is subject to the conditions <*(/)< a, r(Z)>a and E-I^O. Now write G(I)=F(I) p e -\I\ (where / denotes any interval). Thus defined, the function clearly fulfils the conditions: 1 < (?(>(#) < 2e and (a )(x) > a at any point x e E, 2#(Z)>0 for any interval I such that <J(I)<a, r(Z)^a and E>I=Q. By Lemma 11.1 we can therefore determine a cube (?, belonging to one of the nets Q*, so as to have d(Q)<o, \E-Q\>(1 e)*\Q\ and Q(Q)^S m a~ m '2e-\Q\. From the first two of these relations and from conditions 1 and 2, it follows, on account of Lemma 11.8, that <?(#)> 12- m a m + 1 .(l 2 m e)\Q\. Thus 12 - m a m+l .(l 2 m e)^8 n 'a ~ m .2e for every e > 0, and this is clearly impossible. We arrive at a contradiction and this shows that |4|=0, i.e. that F(x)=F(x) for almost ail x for which F(x)> oo. It remains to be shown that the set of the points x at which the derivative F'(x) is infinite, is of measure zero. Suppose then, if possible, that F'(x)=^+oo at each point x of a set M of positive measure. We may clearly assume that there exists a number i?>0 such that F(I) > whenever I is an interval containing points of M and subject to the conditions d(I) <i? and r(/)^2~ m . There- fore, denoting by R any cube which belongs to one of the nets Q* and which satisfies the relations \M -jR| > (1 2~ ( " H ~ 1) ).|#| and 4(fi)< tb we find easily from Lemma 11.8 that F(B) > 2~ 1 -12~ /71 .&.|,R| for every finite number b. We thus again arrive at a contra- diction and this completes the proof. [ 11] Ward's theorems. 139 It should be remarked that for the validity of Lemma 11.1 it is enough to suppose merely that <1 (instead of o ^2~ m ). Similarly in condition (ii) of Lemma 11.8, the inequality r(7)>2~" 1 may be replaced by r(/)^o. The proofs of the lemmas remain essentially the same; we need only observe that if I is an interval whose parameter of regularity is greater than, or equal to 2~ m , and if a < 1 is a positive number, the interval I can always be subdivided into a finite number of non-overlapping sub intervals I,, where ;=1,2, ..., such that r(/,)^a and |Zy|^fc a |J|, where k a is a constant depending only on a. We can now easily see that Theorem 11.15 may be stated in a slightly more general form as follows: any additive function of an interval F is derivable at almost all the points x at which either F^(x) ^ oo or F^(x)< -f-oo, where a is any positive number less than 1. The question whether the condition ^ 1 is necessary here, does not seern to have been solved yet completely. It may how- ever easily be proved (by the method of nets used in the proof of Lemma 11.8) that for any additive function F, the set of the points x for which either F,^(x) = oo or F(i)(x) = -\-oo is of measure zero. For a discussion of these questions, vide the memoir of A. J. Ward [5]. We shall now proceed to prove the second theorem of Ward, in which the ordinary extreme derivates J^(,T) and F(x) of Theorem 1 1.15 are replaced by the strong derivates t\(x) and F^)- It should be remarked however, that we cannot at the same time replace, in the assertion of Theorem 11.15, derivability in the ordinary sense by derivability in the strong sense: in fact, in general, a non-negative function, even when it is absolutely continuous, may yet have a strong upper derivate which is everywhere infinite (see p. 133 above). We shall begin by proving the following lemma which is sim- ilar to Lemma 11.1. (11.16) Lemma. If is an additive function of an interval in lt m and if for some fixed number a we have < Q s (x) < a at every point of a set E of positive outer measure, then given any e > there exists an interval Q such that (11.17) 6(Q)<, r(Q)>2~ m , E-Q*0 and G(Q)<3 m -a.\Q\. Proof. Let us write for brevity y=l/3 w . We may suppose (by replacing, if necessary, the set E by a subset of positive outer measure) that G(I)>0 for every interval I containing points of E, which has diameter less than a positive number a<e. Let x eE be a point of outer density for E and let J~- = [a v ft t ; a 2 , ft 2 ;... ; a ;n , b m ] be an interval containing # such that 140 CHAPTER IV. Derivation of additive functions. <11.18) <$(J)<(r, \E'J\>(1 y)-|7| and G(J)<a-|J|. Let us denote by I the smallest edge of J and by % the positive integer satisfying the inequality (11.19) nj, ^b l a l < (nj+1) I Writing d l =(b l &i)l n u ^ us subdivide the interval J into n t equal non-overlapping subintervals where i 1,2, ... , n v We shall call an interval J t of the first kind if \E-Ji\>(l 3y)-|c7/|, and of the second kind in the opposite case. Denoting by ' a summation over the indices i corresponding / to intervals of the second kind, we see easily, on account of the HI second of the relations (11.18), that J^'3y-|J,-!<y-|J|==^y |J/|. Hence, / /=! if p and q are the number of intervals J/ of the first and second kind respectively, we find 3g<n 1 =p+g. Now let us subdivide the interval J into a finite number of non-overlapping subintervals, in such a manner that each of these is the sum of a certain number of intervals J/ among which exactly one is of the first kind. Since 2q<p, the intervals of this subdivision include some which coincide with certain intervals J/ of the first kind, and their number is at least equal to p g^ttj/3. Thus if we denote their sum by A y we find (11.20) |A|>|J|/3. On the other hand, the figure JQA consists of a finite number of non-overlapping intervals* each of which contains an interval J/ of the first kind, and therefore points of E. Consequently, 0(JQ A) >0, and, on account of (11.18) and (11.20) G(A)<G(J)<a-\J\<3a.\A\. It follows that among the intervals J t of the first kind of which the figure A is formed, there exists one at least, <// say, such that 0(^X3 a- </<,. Let us write, for brevity, aj=aj+(i l)d v b ( ^=a l +i^d l and J (t) = J /o =[a?, &?; a,, fr 2 ; ... ; a m , fr J. By the above, J (1) CV , O(J (i) ) < 3a- 1 J (1) l and, since J (1) coincides with an interval J/ of the first kind, |J0.J (1) !>(1 3y)-|J (1) ,; finally by (11.19), 111] Ward's theorems. 141 If we now operate on J (!) just as we formerly did on J (except that we replace y by 3y, a by 3a and the linear interval [a x , ftj by [a 2 ,6 2 ]), we obtain an interval J (2) =K,&?;a^^;a 3 ,6 3 such that flfJ^KS'-a-IJ^I, E^XlS 2 ?).^ and for j=l,2. Proceeding in this way m times, we obtain after m oper- ations an interval J (m) =[a,&; ajj,ft|j; ... ; a^ftJJC ^ such that #(J (m) )<3 m a.|J (m) |, |^.J (m) |>(l_ a 1 "?)-!/" ! and Z<6 y a y <2i for ?=1,2, ... ,m. It follows that r(J (m) )^2~ /71 , and if we write Q=^j (m} and substitute y=3~ m , we find at once that the interval Q fulfils the conditions (11.17). (11.21) Theorem. If F is an additive function of an interval, we have F'(x)=F s (x)^oo [F'(x)~ F s (x) ^ oo] at almost all the points at which F s (x) > oo [F s (x) < + oo]. Thus, in particular, the function F is derivable in the strong sense at almost all the points at which both the extreme strong derivates F*(x) and F 8 (x) are finite. Proof. Since F(x)^F & (x) holds for all x, the function F is, by Theorem 11.15, derivable (in the ordinary sense) at almost all the points x for which F*(x)> oo, and we have only to show further that at almost all these points F'(x)==F s (x). Suppose there- fore that the set of the points x for which F'(x)>l\(x)> oo is of positive measure. We could then determine a number a > and a set B of positive measure such that F'(x) 1\(#)> a at every point xeB. For brevity, write f=a-3~ (m+l) , and let B p denote the set of the points xeB for which pe <Fs(x)^(p+l)-e. Let p be an integer such that \B PO > 0, and write 0(1) = F(I) p Q e- \I\ (where I is any interval). Since G'(x) >G s (x)+a>a at every point XB PO , we can determine a positive number o and a set ECB Po of positive measure, such that G(Q)>a-\Q\ whenever Q is an interval satisfying the conditions (11.22) t(Q)<, r?)>2-" m and But since 0< G s (x)<2e at every point xeEC^B^, there exists by Lemma 11.16, an interval Q subject to the conditions (11.22) and such that G(Q)<3 m -2e-\Q\<a-\Q\. We thus arrive at a contradiction and this proves the theorem. 142 CHAPTER IV. Derivation of additive functions. *$12 A theorem of Hardy-Little wood* The theorem of Jessen, Marcinkiewicz and Zygmund concerning strong derivation of indefinite integrals, which was mentioned in 10, p. 133, is connected with an important inequality due to G. H. Hardy and J. E. Littlewood [2]. This inequality, which was established in connection with certain problems of the theory of trigonometrical series, thus obtains a new and interesting application. We reproduce in this the elegant proof given by F. Riesz [5] (cf. also A. Zygmund [I, pp. 241 245]) for this inequality. Although simpler than the other proofs, it requires nevertheless some rather delicate considerations. Certain parts of the argument have been touched up in accordance with suggestions communicated to the author by Zygmund. The reasonings of this concern functions of a real variable. (12.1) F. Riesx's lemma. Let F(x) be a continuous function on an interval [a, b] and Jc a finite number. Let E be the set of the points x, interior to the interval [a, ft], for each of which the inequality F(x) F(u)>k-(x u) is fulfilled by at least one point u subject to a < u < x. Then the set E is either empty, or else expressible as the sum of a sequence {(a n , b n )} of open non-overlapping intervals such thai Proof. By subtracting from F(x) the linear function lex, we may suppose that fc=0. Then E is the set of all the points x of the open interval (a, ft), for each of which there exists a point u such that F(u) <F(x) and a < u < x. Since the function F is continuous, the set E is clearly open, and, unless empty, it is therefore ex- pressible as the sum of a sequence {(a nj b n )} of non-overlapping open intervals. We have to prove F(a n )^F(b n ) for each n. To see this, let us fix an index n and suppose that F(a n ) >F(b n ). Let h be any number such that (12.2) F(a n )>h>F(b n ), and let X Q be the lower bound of the points x of the interval [a n , b n ~\ for which F(x)=h. By (12.2) the point x belongs to the open interval (a n , ft n ), and so to the set E] thus there exists a point y such that F(y) <F(x )=h <F(a n ) and a < y < # . This last relation implies a < y < a, since, by (12.2) and by the definition of the point [$12] A theorem of Hardy-Littlewood. 143 o? , the inequality F(y)<h cannot hold for any y of the interval [a w ,0 ]. Thus F(y)<F(a n ) and a<y<a n , and consequently a n eE; but this is' clearly contradictory, since a n is an end-point of one of the non-overlapping open intervals which constitute the set E. Besides the results treated in this , many other applications of Lemma 12.1 are given by F. Riesz [6; 7; 8], particularly in the theory of derivation of func- tions of a real variable. Cf. also S. Izumi [1], The lemma might also have been used in the considerations of 9 (instead of appealing to the theorems of 8 on rectifiable curves). To shorten our notations we shall restrict ourselves in the rest of this to functions defined in the open interval (0, 1); and we shall agree to write B[/ > a] for E[/(#) > a; < x < 1]. The symbols X E[/^a], E[ft>/>a] and so on, will have similar meanings. Two measurable functions g and h in (0, 1 ) will be called (in accordance with the terminology of F. Eiesz) equi-measurable if |E[#> a]|=|E[A> a]| for every finite number a. We see at once that we then also have E\[b > g ^ a]|=|E[6> h^a]\, etc. (12.3) If two non-negative measurable functions g and h in the in- terval (0, 1) are equi-measurable, their definite integrals over this in- terval are equal. To see this, let us associate with the function g a non-decreasing sequence {g n } of simple functions by writing g n (x)=(k l)/2" when (fc l)/2"<0(o?)<fc/2' 1 and fc=l,2,...,2 n -n, and n (#)=nwhen g(x)^n. Similarly with g replaced by ft, we define the sequence {h n } con- verging to the function h. If we calculate directly the integrals of the functions g n and h n over (0, 1) by formula 10.1 of Chap. I, p. 20, we see at once from the fact that the given functions g and h n->oo, are equi-measurable that fg n (oo)dx=fh it (x)dx. Making i this gives fg(x)dx=fh(x)dx as asserted. o o If / is a continuous function in (0,1) which is not constant on any set of positive measure, and if m and M denote the lower and upper bound of / respectively, the function 0(j/)=|E[/>y]| is evidently continuous and decreases from 1 to in the open interval 144 CHAPTER IV. Derivation of additive functions. (m, M ). Its inverse function is therefore continuous and decreasing in (0, 1) and, as we easily verify, equi-measurable with the given function. We sfrall extend this process with suitable modifications, to arbitrary measurable functions finite almost everywhere in (0, 1). With any such a function /(a?), we associate the function f(x) de- fined for each x of (0, 1) as the upper bound of the numbers y for which |E[/>t/]|>#. The function f(x) is clearly finite and non- increasing in (0, 1). To show that this function is equi-measurable with f(x), let i/o be any finite number, and let X Q denote the upper bound of the set E[/ rt > y ], or else # if this set is empty. Then since |E[/"> y^]\ = # , it has to be proved that |E[/> t/ ]| = # . We have, in the first place, /"(#<>+) ^y ^ or ever y f >0 (provided, of course, that # +e<l), so that |E[/> 2/o+ f ]|^#o+ f > and therefore |E[/ > y ]| ^ #o On the other hand, f(x Q e) > y for every C>0 (provided that x C>0), so that |E[/>y ]|>tf e > whence |E[/> y ]|^o, and finaUy |E[/> y ]|=# ==E[f > t/ ]- We shall further define, in connection with any summable function /(#), three functions f l (x), f*(x) and f(x). At any point x of (0, 1) we shall denote by f l (x) the upper bound of the mean values of / on the intervals (u,x) contained in (0, a?), i.e. the 1 * upper bound of the numbers - lf(t)dt for 0<w<#. Simi- x w J u a 1 r larly, f '(x) will denote the upper bound of the means - / f(t) dt \) - X ' JC for a?<v<l. Finally, f(x) will denote the larger of the two num- bers f\x) and f*(x), or what comes to the same, the upper bound 1 * of the means _ Jf(t)dt where u and v are subject to the u condition (12.4) Lemma. If f(x) is a non-negative measurable function in the 'open interval (0, 1) and if E is a set contained in this interval, then \E\ ff(x)dx<ff(x)dx. E [12] A theorem of Hardy-Littlewood 145 Proof. Let f 1 be the function equal to / on the set E and to elsewhere. We evidently have f"(x)^f(x) at each point x of the in- terval (0,1). Furthermore /"(#) = as soon as x>\E\. Therefore on 1 1 \E\ account of (12.3) we find ff(x)dx=ff l (x)dx=ff(x)dx^ff a (x)dx. 6 o o (12.5) Lemma. If f(jc) is a non-negative summable function in the interval (0,1), then for each point x of the interval, we have Proof. Let X Q be any point in (0,1), let yQ=f* ia (x ), and let denote an arbitrary positive number. We write A = E[f l(t >y a e] and B=E[/ I?I >// ej. Then since the function /*'" is non-increasing, we have \A\^x Q and therefore, remembering that the functions f* tt and /'*' are equi-measurable, |B|=|A|^a: . Now B is the set of the points x for each of which there exists X a point ?/ subject to the conditions If(t)dt>(y e)-(x ?/) and ;.r. Therefore, applying F. Riesz's Lemma 12.1 to the in- definite integral of /, we find easily that B is an open set and that lf(t)(lt^?(!/Qf)'\B\. It follows by Lemma 12.4 that B (12.6) ?/<> * <ril /W) de ^T15i //"(*)*- \J*\B 1^1 6 Now since \B\^x and since the function f" is non-increasing, the last term of (12.6) cannot exceed ff(t)dtj and since t is an A',, arbitrary positive number, we must have f lCC (%o)=yo^ / f"(t)dt. This completes the proof. (12.7) Theorem of Hardy-Littlcn'ootl. If f(x) is a non-neyatire wminable function in (0, 1) and e in a positive number, then (12.8) o' (i where A and B are constant* depending on e, but not on /. 146 CHAPTER IV. Derivation of additive functions. Proof. We first evaluate the integral of f l over (0, 1). Ac- cording to (12.3) and Lemma 12.5 we have o o o o In virtue of Fubini's theorem the last member of this inequality is the surface integral of the function f(y)jx over the triangle 0<:#^1, O^y <: x. Therefore inverting the order of integration in this member, we find (12.9) ff(x) dx </ [ / ^]f(y) dy= (f(y)> |log y | dy. o o L^ J 5 y Let now *?<! be a positive number such that [\logy\lYydy <*/2. 6 Let us denote by E l the set of the points y of the interval (0, rj) at which f(y) < 1/1% and by jE 2 , the set of the remaining points of this interval. We find (12.10) /f(y)-|logy| dy<f dy+2ff(y) : logf(y) dy+ 1 +Nf ll'J f(v) dy < 2J f(y)-log+ f(y) dy +|log i?|- J /"(y) dy+e/2. Y] Further, since the functions / and /" are equi-measurable, so are the functions /-log^/ and /"-log^/", and it therefore follows from (12.9) and (12.10) that (12.11) A similar inequality clearly holds when on the left-hand side of (12.11) f l is replaced by f* 9 and on adding the two inequalities we find Jf(x) dx <//'(#) dx+jf\x) dx < ff(x) log+ j(x) 000 1 + |21og^|-y/(a?)da?+e; this gives ( 12.8) with A = 4 and with =2|logi?|. f13] Strong derivation of the indefinite integral. 147 * $ 13. Strong derivation of the indefinite integral. We proceed to prove the theorem of Jessen, Marcinkiewicz and Zygmund. We shall give the proof for the case of the plane; its extension to spaces H m of any number of dimensions (cf . 10, p. 133) presents no fresh difficulties and is effected by means of the well-known inequality of Jensen. We shall begin with some auxiliary remarks. Suppose given a non -negative function /(a?, y) summable over the open square J =(0, 1; 0, 1). By Fubini's theorem, the function j(x,y) is sum- mable in x over (0, 1) for almost all y of (0, 1). Denote by H the set of these values of y. For any y H and for any x of the interval (0, 1), we shall denote (cf. 12, p. 144) by f(x,y) the upper bound 1 '' of the mean f(t,v)dt for 0<w<,r<^<l: and whenever v u-J ' J J/ yfC/7, we .shall write, for definiteriess, /'V? y) = identically in x. We shall prove that the function /'V? y) thus associated with any function f(x, y) which is summable over the open square J = (0, 1 ; 0,1 ), is measurable. For this purpose, let a and b denote two positive numbers, .v-f ft and write 0,i.&(j?, y)= ff(t, y)dt when ycH and 0<^- a<x i &<1, \ <i and J/,,.A(JP, #) = elsewhere in J . We shall begin by showing that each of the functions </ fli &(.r, //) is measurable. By Lusin's theorem, or more directly by the theorem of Yitali-Caratheodory (Chap. Ill, 7), the function / is equal almost everywhere to the limit of a non- decreasing sequence J/ (/I) ) of non-negative, bounded, upper semi-contin- .v-t- b uous functions. Xow, let us put y ( "\ t (x* y)= J f (n) (t, y) <lt when x (i ^ ,/; a < jc -f- b ^ 1, and y*"\(Xj y)=0 elsewhere. As is easy to show (e. g. by means of Theorem 12.11, Chap. 1), each of the functions y^"\(x 9 y) is then also upper semi-continuous, and since, as we readily see, g fl 6 (j?, y)]imy ( f ^ h (x,y) almost everywhere, the function g nt b(x^y) n is measurable. Finally, with the same notation as above, if [u f) ] is the sequence of rational numbers of the interval (0, 1) we have /"(.r, y) = upper bound g u tU (#, y) t (Up w (y ) 148 CHAPTER IV. Derivation of additive functions. at any point (#, y) of J . Thus the function f(x, y) is also measurable, and this proves our assertion. (13.1) Theorem. If f(x,y) is a measurable function in the plane U 2 and if the function f log+ |/| is summable, then the indefinite integral of f is almost everywhere derivable in the strong sense. Proof. Clearly we need only consider the function / in the open square J =(0, 1; 0, 1); and we may also suppose that this function is non-negative. We write g n (x,y) = f(x,y) wherever f(x, y)<, and g n (x,y)=n wherever f(x,y)^>n; we write further h n (x, y) = f(x, y) g n (x,y) and we denote by o an arbitrary positive number. The functions jf n (Xj y) are measurable and non-negative; so that by Theorem 12.7 of Hardy-Littlewood, (13.2) . ^AJ fh n (x,y)-log+h n (x,y)dxdy+BJ fh n (x, ',, ./ where A and B are finite constants depending only on a. And since the integrals on the right-hand side of (13.2) tend to as w->oo, there exists a positive integer N such that the left-hand side of (13.2) becomes less than a 2 for n = N. Therefore, writing for brevity h(x, y)=h N (x, y) and g(x, y)=g N (x, y)j we have (13.3) ffh(x,y)dxdy<o*, /o so that in particular the function ti*(x, a/), besides being measurable and non-negative, is summable on J . Now denote by E the set of the points (r , y Q ) of J such that i y 1 I'h ? (x ,t)dt< + oo, and 2 the indefinite integral jh*(x ,t)dt 6 o has at the point y = y the derivative h^(x QJ y Q ) with respect to y. Since, by Theorem 6.3, condition 2 is fulfilled for almost all ?/ of (0, 1 ) provided that condition 1 is satisfied, it follows at once from Fubini's theorem in the form (8.6), Chap. Ill (cf. also Theo- rem 6.7, Chap. Ill) that |B|=|J |=1- Let us write F, H and for the indefinite integrals of the functions /, h and gf, respectively, in J . Let (X QJ y ) be a point of the. set E and I = [> ^uXg + u^y^ 1\, y Q + v 2 \ any interval con- taining (x , y ) and contained in f/ . We have [ 14] Symmetrical derivates. 149 '_ U 2 whence making <5(Z)->0 we obtain //^(# , y )^^( x o^ yo)- Thus, since (# , // ) is an arbitrary point of the set E(2J Q f outer measure 1, and since the extreme derivate H k (x , y ) is measurable (cf. Theo- rem 4.2), it follows from (13.3) that 0< Hs(#, y)^ H.(x,y)^a at every point (x, y) of J , except at most a set of measure less than a. On the other hand, since the function g g N is bounded, its in- definite integral G is, by Theorem 10.7, derivable in the strong sense almost everywhere. Therefore F*(x,y) F K (Xjy)^o at all but a subset of measure a of the points of J ; and so finally, since o is an arbitrary positive number, F k (x, y)=F*(x, y) almost every- where in J , which completes the proof. By Ward's Theorem 11.21, to prove that the non-negative function F is almost everywhere derivable in the strong sense, it is enough to show that ^(^X-f almost everywhere. Hence by using Theorem 11.21, the proof of Theorem 13.1 might be slightly shortened. *14. Symmetrical derivates. If <is an additive function of a set in a space JK mj we shall denote by D^ m 0(x) the upper, and by p s x rn 0(j?) the lower, symmetrical derivate of at a point #, these being defined respectively as the upper, and as the lower, limit of the ratio &(S)I\S\ where 8 represents a closed sphere of centre x and of radius tending_to zero. It is obvious that, for any point x whatsoever D0(a?)^D 8 y m *(^)^p S x m 0(a?)^p0(a?). Following A. J. Ward [5], we shall establish a decomposition theorem in terms of symmetrical derivates, which is similar to Theorem 9.6. We shall begin by the following u covering theorem": (14.1) Theorem. If is an additive function of a set in R m and E a bounded set measurable (93), contained in an open set <?, then for any e > there exists in an enumerable sequence of closed sphe- res {Sft} such that (i) the centre of each 8k belongs to E and the ra- dius is less than e, (ii) $ r $/ = whenever i=H, and (ui) the spheres 3k cover together the whole of the set J5, with the possible exception of a subset on which the function $ vanishes identically. 150 CHAPTER IV. Derivation of additive functions. Proof. We can clearly assume (by replacing, if necessary, the function $ by its absolute variation) that the function $ is monotone non-negative. a) We shall first prove that, with the hypotheses of the theorem, there always exists in G a finite system of equal sphe- res {S k } which sasisfy the conditions (i) and (ii) and cover the set E except perhaps for a set T(2 E such that (14.2) 0(T) < (1 l/4' n+1 w m )- $(E). To see this, let A be a subset of E, measurable (53), such that 0(A)^$(E) and 0(A,C(?)>0. Let n be a positive in- teger such that m/n Q <(>(A, CG) and m/w <e. Denote by ty the net in the space JR, fJ , which consists of the cubes of the form [p,/n , (p,+l)/n ; p 2 /n , (p 2 +l)/n ;...,p m /n , (p m +l) n ] where p v p 2 , ..., p m are arbitrary integers. We can clearly sub- divide the net ^ into (4w) m families of cubes, ^u^^wm)" 1 say, such that the distance between any two cubes belonging to the same family is not less than (4m l)/w . Denote, for each fc=l,2, ...,(4w)" f , by A* the part of the set A covered by the cubes of the family ty k . Then there exists a positive integer k Q ^(m) m such that (14.3) Now let PI, P-2, ..., P r be those cubes of ^ which contain points of Ak u . With each P/ we can associate a closed sphere S/, of radius m/n , whose centre belongs to A/^-P/. The system of spheres S\jSz,...)Sr thus defined is contained in G and clearly satisfies the conditions (i) and (ii) of the theorem. Again, since P/C$/ for every i = l, 2, ..., r, the spheres $, cover the whole of the set E with the possible exception of the points of the set T=E~-A k ^ which, in virtue of (14.3), fulfils the condition (14.2) b) We now pass on to the proof of the assertion of the theorem. By what has already been proved, we call define by induction a sequence {Q n } n ^ { 2 of finite systems of closed spheres with centres in E and radii less than e, subject to the following two conditions: 1 If B denotes the empty set and B/,, for n^l, the sum of the spheres belonging to Si + 62 + + 6/i, then the system <5 n -fi, where n^O, consists of a finite number of closed spheres, contained in the open set OB n , no two of which have common points; 2 0(E-B n+l )^(l-h m )^(E^B H ) where A m -l/4 m4 ' 1 m m and n=0, 1, ... N\>w, arranging the spheres belonging to the family l14] Symmetrical derivates. 151 ... + S,,+... in a sequence {#/}, we see at once that the latter fulfils conditions (i) and (ii) of the theorem. On the other hand, by 2 we have $(E-- B n )^(l- h m ) n -<D(E) for each n; whence, denoting by B the sum of all the spheres $/, it follows that <P(J57 JB)=0, which establishes condition (iii) and completes the proof. Theorem 14.1 may be established in a slightly more general form: Given a bounded set E measurable (33), a sequence of positive numbers {r n } converging to and a family of closed sets 31, suppose that with each point x of E there are associated two finite numbers o=a(a?), NN(x), and a sequence {A n (x)} of sets (91) such that S(x; r n )C^n(x)C^(x; <*r n ) for n^N(x). Then, for any sequence {$n} of additive functions of a set, we can extract from ty a sequence of sets {AtAn^x^} such that (i) xieE for i=l,2, ..., (ii) AfAj=*Q whenever i-^j, and (iii) the sets At cover the whole of the set E, with the exception at most of a set of measure zero on which all the functions $ n vanish identically. (14.4) Lemma. If is an additive function of a set in R mj and if p sym <(#)> at each point x of a bounded set X measurable (53), then 0(X)^*0. Proof. Let us denote, for every positive integer n, by X n the set of the points xeX such that $ (8)^0 whenever Sis a closed sphere of centre x and radius less than l[n. Bach set X n is evidently meas- urable (93), in fact closed in X. Hence, for any e>0, we can associate with each X n an open set G n ^)X n such that W(<P; G n X n )^f (cf. Theorems 6.9 and 6.10, Chap. III). Next, keeping n fixed for the moment, we can (on account of Theorem 14.1) define in G n a sequence {$*} of closed spheres with centres in X n and radii less than 1/n, such that (i) $/$/=<) whenever i^pj, and (ii) the sphe- res 8 k cover the whole of the set X n with the exception at most of a set T on which the function <P vanishes identically. Sinde for every fc, we find by (i) and (ii) that $(X n ) ^ ) + W(ViG n X n )]^e. Hence, as X=*timX n and e n is an arbitrary positive number, it follows that (P(JT)^O, which completes the proof. (14.5). Theorem. If is an additive function of a set in R mj <md if AM denotes the set of x at which one at least of the derivates D 8ym <P(a?) and jD 8ym 0(a?) is infinite, then for any bounded set X measurable (35), we have (14.6) 0(X) Consequently, if D sym #(#)>-- oo at every point x and at almost every point x of a bounded set X measurable (33), then 152 CHAPTER IV. Derivation of additive functions. Proof. We firstly remark that if oo<D sxm 0(,r) at each point x of a bounded set Q measurable (5?) and of measure zero, then 0($)^0. In fact, denoting for each positive integer n by Q n the set of the points x of Q at which >Kl), un 0(#), and writing 0,,(A r )=0(.Y) + tt-LY|, we obtain P,, m 0,,(,c)>0 at every point xeQ n . Hence, by Lemma 14.4, we must have 0(Q W )=0/,(#,,)^0, arid making n~>oo we find 0(Q)^0. By symmetry we also have 0(^)^0 when- ever Q is a bounded set measurable (93) of measure zero, such that D0( .#)<-{- oo at each point x of Q. We pass on to the proof of formula (14.6). By Theorem 7.3, there exists a set A, measurable ($) and of measure zero, such that the relation (14.7) &(X)=0( x holds whenever X is a bounded set measurable (33). Since the set Aoo is of measure zero, we see at once from the equation (14.7) that the function must vanish identically for all the subsets of A^A, which are bounded and measurable (53). On the other hand, by what has just been proved, the function vanishes also for all subsets of A An. Hence the set A& may be taken in place of the set A in (14.7) and this gives (14.6). Finally, if D S x m 0(#)> oo at every point # of a bounded set X measurable (95), then 0(X-A OQ )'^Q and the second part of the theorem follows at once from the first. Let us mention the following consequence of Theorem 14.5: // at each point x both the symmetrical derivates oj a given, additive function of a set are finite, the latter is absolutely continuous. For ordinary derivates the corresponding pro- position has long been known (cf. II . Lebesgue (5, p. 423 j) and is moreover included in Theorem 15.7 of this chapter, an well as in Theorem 2.1 of Chap. VI. *15. Derivation In abstract spaces. With certain hypo- theses, a process of derivation may be defined for additive func- tions of a set in any separable metrical space, and for such a process, theorems similar to those of 7 and 9 may be established. (15.1) Lemma. If is an additive function of a set ($>) on a metrical space Jlf, then given any set X measurable () and any e>0, there exists an open set G such that (15.2) W(0; X)<e and W(0; X 0)<e. l 15] Derivation in abstract spaces. 153 Proof. Let ^ denote the class of the sets A' measurable (93) for each of which there exists, however we choose e>0, an open set satisfying the relations (15.2). Since any closed set F is the limit of a descending sequence of open sets, we observe easily (of. Theorems 5.1 and 6.4, Chap. I) that there exists for each *:>0 an open set G^)F such that W(0; GF)<e. The class <B thus includes all closed sets; to prove that 93=93 , it suffices, therefore, to show that the class 93 is additive. To do this, we choose e>0 and denote by X the sum of a se- quence (AVf/i-i.2,... of sets (93 ). To each set X n there corresponds an open set G n such that W(#; G n X n )<e/ i 2" and W(0; X n G /l )<e/2 /I . Writing G=^G fl , we clearly find that the inequalities (15.2) are n satisfied. Therefore Ae93 . Again, suppose that e>0 and that JF=Cr, where Yf93 . There will then exist an open set H such that W(0; Y #)<e/2 and W(0; H Y)<e. Consequently writing P=CH, we find that (15.3) W(</>; P X)<e/2 and W(0; X P)<e. But since the set P is closed, there exists an open set G such that 03 P and W(0; G P)<e/2; and this implies, on account of (15.3), the inequalities (15.2) and so completes the proof. We shall call net in a metrical space Jf any finite or enumer- able family of sets measurable (53) no two of which have common points and which together cover the space M. The sets constitut- ing a net will be called its meshes. A sequence (2R,,) of nets will be termed regular, if each mesh of 3Jl n +i (where n>0) is contained in a mesh of 3R ; , and if further 4(9M,,)->0 as n ->oo (where d(9Jl,,) denotes the characteristic number of 2R,,; cf. Chap. II, p. 40). It is easy to see that in order that there exist a regular sequence of nets in a metrical space, it is necessary and sufficient that this space be separable. In the rest of this we shall keep fixed a separable metrical space J/and we shall suppose given in J/ a regular sequence 3J? = (3R,,} of nets and a measure // which is defined for the sets measurable (93) and which is subject to the condition //(Jf)<+oo. Let be an additive function of a set (93) on M. For x e If, where M is any mesh of a net 9JI,,, let us write 0(M)lfji(M) when ^(Jf)=J=0, + oo W h en p(M) = i) and oo when //(3/)=0 and 154 CHAPTER IV. Derivation of additive functions. The functions d n (x) are thus defined on^the whole space M and are measurable (33). Let us write (p, 2R)D<(#) = lim sup d n (x). _ 71 The number (//, 3K)D0(a;) thus defined wiU be called upper derivate of the function at the point x with respect to the measure p and the regular sequence of nets 3JI. Considered as a function of a?, this upper derivate is clearly measurable (93). Similarly we define thejower derivate (//, 3Jl)I)0(x). If at a point x the two numbers (//, 2R)D<P(a?) and (/A, 3Jt)I)0(#) are equal, their common value will be written (/*, 3R)D0(#) and called derivative of the function at x with respect to the measure JLI and the regular sequence of nets 901. For the rest of this , a measure fi and a regular sequence of nets 3R will be kept fixed in the space M. (15.4) Lemma. Let be an additive function of a set (53) on the space M. Then (i) if the inequality (//,3R)D0(#)^i, where k is a finite number, holds at every point x of a set A measurable (93), we have 0(A)^k-ju,(A)i (ii) if at each point x of a set B measurable (33) and of measure (JA) zero, the derivative (//,9Jt)D0(#) either does not exist or else exists and is finite, Proof, re (i). By subtracting from the function the function fc-/i, we may assume that fc=0. Let e be any positive number. By Lemma 15.1 there exists an open set such that (15.5) W(0; A)<e and W(0;AG)<e. Let aRj be the set of the meshes M of the net 90^ such that (15.6) MCG and and generally, for n>l, let $}+! be the set of the meshes M of the net 9K n +i which fulfil the conditions (15.6) and are not contained in any of the meshes of OTj+SWj+.-.+^n. By arranging the sets belonging to 9KJ+SWJ+...+9KH+... in a sequence {M *}, we have <p(M k )>e-fii(M k ) for fc=l, 2, ... , and A-GCZMk. Since the sets M k are measurable (93) and no two of them have common points, it therefore follows from (15.5) that 4>(A) = 0(A.G)+0(A and so that [15] Derivation in abstract spaces. 155 re (ii). Denote, for any positive integer n, by B n the set of the points x of B at which T)0(x)^ n. On account of (i) we have 0(B n )^ U'ju,(B n ) = Q for each n, and, since B = lim B n , this gives 0(J5)>0. By symmetry 0(J3)<0, and so finally 0(B) = 0. (15.7) Theorem. If is a function of a set (53), wAt'cA t* additive on the space JJT, Me derivative (//,9pl)D0(a?) ozrcste almost everywhere and is integrable (93, ^) on 3J; moreover, if E^.^ and E-& denote the sets of the points at which (//, 9H)D0(o?)= + oo and (//, 5)t)D<P(#) = oo respectively, we have (15.8) , X and (15.9) for every set X measurable (95). Proof. By Theorem 14.6, Chap. I, there exist a function of a point / integrable (95, //) on M and an additive function of a set & singular (93, /*) on 3JT such that (15.10) 0(X) = S(X) + ff(x)dii(x) for every set Xe 93. Let E be a set measurable (93) such that //() = and that the function (9 vanishes identically on CE. Writing, for brevity, D, I) and I) in place of 0*,2R)D, (/*, 9R)D and (^,9R)D respectively, let us denote for any pair of integers n>0 and fc, by P,,,/? the set of the points x at which D0(x)^(k+l)/n>k/n^f(x). If we substitute Pn t k-CE for JT in (15.10), we find on account of Lemma 15. 4 (i) that ^^ H and so that /i(P n ^)=^(P nfjk .CJE?) = 0. Therefore D&(x)*^f(x) at almost all points ^. By symmetry D<P(x)^f(x) must also hold almost everywhere in M. Therefore the derivative D<P(x) exists and equals f(x) at almost all the points x of M, and the identity (15.10) takes the form (15.11) 0(X) = 9(X)+fD<Pd ] u=0(E.X)+fD0d/* for every set X X Moreover, since D4>(x)f (o?)^oo almost everywhere, the set jE is of measure (//) zero, and it follows directly from (15.11) that the function <P vanishes identically on the set (jE+oo+-E-<x>) E. 156 CHAPTER IV. Derivation of additive functions. On the other hand, by Lemma 15.4 (ii), vanishes identically on EiE+co+E-ao). Therefore in (15.1 1) the set E may be replaced by the set E+oo+E _, and the relation (15.11) becomes the required formula (15.8). Finally, since by Lemma 15.4 (i) the function # is non-negative for the subsets (93) of E+*> and non-positive for the subsets ($) of E. , formula (15.9) follows at once from for- mula (15.8). Let us mention specially the following corollary of Theorem 15.7: (15.12) Theorem* Suppose given in the space M two regular sequences of nets $1 and Sp, and, as before, a measure ,n defined for the sets ( S B) and subject to the condition /*(>/)< -f oo. Then for every function $ of a set ("B), which is additive on >/, we have almost everywhere (,, $l)D$(x) (ft, Sp)D#(jr); moreover, if E denotes the set of the points x at which either one at least of the derivatives (/*, 9i)D$() and (//, ^J)D^(-r) does not exist, or else both exist but have different values, then the function $ vanishes identically on E, i.e. W(<P;!)=0. In fact, if we write, for brevity, D! and D 2 in place of (//, 91 )D and (,, $)D respectively, and if we denote by # the function of singularities of 0, we have by the previous theorem for every set X measurable ($3). Equating the two integrals which occur in this relation, we obtain almost everywhere D 1 <I>(x) D 2 0(x). Now the set E of the points at which this relation does not hold, may be expressed as the sum of three sets A l9 A 2 and A 3 , where A L is the set of the points xe at which one at least of the derivatives D^(x) and D t $(x) does not exist, or else exists and is finite, A 2 the set of the points x at which J) l (x)~-\-oo and D 2 $(x)^-oo, and A 3 the set of the points x at which D l ^(x) = -oo and D 2 Q(x)=+oo. It follows directly from Lemma 15.4 (ii) that the function # vanishes identically on A r In the same way, it follows from part (i) of this lemma that we have simultaneously 0(-Y)_-0 and $(Z)^0, and so 4>(X) = 0, for every subset X measurable ( s #) of A 2 or of A 3 . Consequently W(#;/tf)=0, and this completes the proof. Theorem 15.7, which corresponds, to a certain extent, to Theorem 9.6, was first proved by Oh. J. de la Valle'e Poussin [1; cf. also I, p. 103] for derivation with respect to the Lebesgue measure, and with respect to the regular sequences of nets of half open intervals in Euclidean spaces. Strictly, the Lebes- gue measure does not fulfil the condition which we laid down for the measure p t since Euclidean space has infinite Lebesgue measure. Nevertheless it is easy to see that for the validity of Theorem 15.7 (as well as for that of the other pro- positions of this ) it suffices to suppose only that the meshes of the nets considered have finite measure. For the derivation of additive functions of a set in abstract spaces, see also R. de Possel [1]. [16] Torus space. 157 * $ 16. Torus space. As an example and an application of the results of the preceding , we shall discuss in this a metrical space which, from the point of view of the theory of measure and inte- gration, may be considered as one of the nearest generalizations of Euclidean spaces. This space, called torus space of an in- finite number of dimensions, occurs in a more or less ex- plicit form in the important researches of H. Steinhaus [2], of P. J. Daniell [2; 3], and of other authors, in connection with certain problems of probability; but the first systematic study of this space is due to B. Jess en \2]. Following .lessen, we shall call torus space Q (l > the metrical space whose elements are the infinite sequences of real numbers g=(x lj # 2 , ... ,#, .,.) where 0^# n <J for w = l,2, ..., the distance (, fj) of two points f = (x ly x 2 , ... , a?,,, ...) and 17= (ft, y ... , #/ ...) in Q> being defined by the formula Q(, rj)=2\y n x n \/2 n . By (? m , n where m is any positive integer, we shall denote the half open cube [0, 1; 0, 1; ...; 0, 1) in the Euclidean space Jt m . If =(x ly x 2 , ... ,#, ...) is a point of Q^, we shall denote, for any positive integer m, by m the point (x x 2 , ..., x m ) of Q mj and by f^ the point (x m +\> tf m +2, ) of Qu, and we shall write f=(f m , f^). According to this notation, (,7?) is a point of Q^ whenever SfQm (where m is any positive integer) and r}Qi. So that, if A(2Q m arid #C (?<>> the set A xB (cf. Chap. Ill, 8, 9) lies in the space ( w ; and in particular Q m xQuF=Qu>. We shall call closed interval, or simply interval, in the space Q M , any set of the form / x <?o>, where / is a closed subinterval of Q m for some value of m 1, 2, Similarly, taking I to be an interval which is half open (on the left or on the right) in Q mj we define in the space Q< the half open intervals (on the left or on the right). Every (closed) interval J in Q^ has only one expression of the form /x Q M where / is an interval in a space Q m . (It is to be re- marked that the space Q^ itself is not a closed interval in the sense of the definitions given above.) By the volume of the interval JI x Q M we shall mean the volume of the interval / in Qmd^m (cf. Chap. Ill, 2). Just as in Euclidean spaces, the volume of an interval J in Q will be denoted by \J\ or L W (J). Again, as in Euclidean spaces (cf. Chap. Ill, 5), we shall extend the notion of volume in the space Qu by defining for every set E in this space the outer measure L*(E) of the set E as the lower bound of the sums J k \ where (JV, is k any sequence of intervals such that EC.2jJ*- Thus defined, the 158 CHAPTER IV. Derivation of additive functions. outer measure evidently fulfils the three conditions of Carath^odory (cf. Chap. II, 4) and determines, first the class of sets measurable (L^), and then the class of functions measurable (2L*J. For brevity, the sets and the functions belonging respectively to these classes, will simply be termed measurable. Also by the measure of a set E in the space Q u we shall always mean its measure (L*). It is easily shown, with the help of BorePs Covering Theorem, that the measure of any closed interval coincides with its volume (cf. Chap. Ill, 5, p. 65), so that we can, without ambiguity, write \E\ or L^(E) (omitting the asterisk) to denote the outer measure of any set E in Q^. We also see that the boundary of any closed interval is of measure zero. Finally, we remark that the whole space Q& is of measure 1. We shall now define in Q t> > a regular sequence of nets (cf. 15, p. 153) of intervals half open on the right. We shall, in fact, denote for any positive integer w, by Q (/n) the finite system of 2" r intervals half open on the right whore the fc/ are arbitrary non-negative integers less than 2. We see at once that each system Q (m) is a net in Q M . To see that the sequence of these nets is regular, we observe in the first place that each interval of Q (mH) is contained in one of the intervals of Q ( " J) . Oil the other hand, no interval of the net Q (/;l) can have a diameter exceeding the number J4/2 mf *+j;i/2*<l/2 m ~" 1 , so that k- 1 A-m-fl the characteristic number J(Q (/n) ) of the net Q (m) tends to zero as ra->oo. If x and y are two real numbers, x + y will denote the number v+y [+y]i where, as usual, [x + y] stands for the largest integer not exceeding jc + y. If f =(# j? 2 , ... , a?.,, ...) and *?=(# # 2 , .,//, ) are two points of Q M , we shall write +// for ( The point f+*7 clearly belongs to Q t< >. We shall call translation by the vector a, where a is a point of Q to , the transformation which makes correspond to each point f of Q t * the point $+a. The translation by the vector a will be termed of order w, if all, except at most the first m, coordinates of a vanish. A function / in Q will be termed cylindrical of order m, if /(f) does not depend on the first m coordinates of the point , i. e. if 1(16] Torus space. 159 /()=/( + a ) identically in , for every point a whose coordinates, except perhaps the first m, all vanish. A set E in Qu will be termed cylindrical of order m, if its characteristic function is so, or, what amounts to the same, if E=Q m xA where A is a set in Q*. (16.1) Theorem. A function which is measurable on Q<* and cylindrical of every finite order, is constant almost everywhere, i. e. /(f)=c for almost all points of Q^, where c is a constant. Proof. Suppose first that the function / is bounded, and therefore integrable, on Q<*. Denoting by the indefinite integral of /, let us define, for each value of m and for each mesh Q of the net Q (m) , f m \)=&(Q)/L(Q) whenever Q. For every pair of meshes Q l and Q 2 of the same net Q (m) , there always exists a trans- lation of order m which transforms Q l into Q 2 ; therefore, since the function / is cylindrical of order m, it follows that &(Qi)=&(Qi)i and since further L ft) (Q 1 )=L w (Q 2 ), each of the functions / (m) () is constant on Q M . On the other hand, we deduce from Theorem 15.7 that /( ) = lim / (m) (f ) almost everywhere in # w , i. e. that the function m is almost everywhere identical with a constant. Now let / be any measurable function which is cylindrical of every finite order. Let us write //,()= /(f) when \f()\^n and f(g)=n when |/()|>n. Each of the functions /() is bounded and cylindrical of every finite order, so that by what has just been proved, each of these functions is constant almost everywhere. Therefore the same is true of the function /(!) = lim /( ) and this completes the proof. n The fundamental properties of our measure in the space Q* may be established by methods similar to those used in Euclidean spaces. To illustrate this, let us enumerate some of these properties. Given any measurable set E and any e>0, there exists a closed set F and an open set O such that FC E G G and such that \G E\<e and \EF\<e (cf. Theorem 6.6, Chap. III). From this we may deduce next Lusin's theorem (cf. Theorem 7.1, Chap. Ill): If f is a finite function measurable on a set E, there exists for each e>0, a closed set FC.E such that the junction f is continuous on F and that \E F\<, and its immediate corollary: any function which is measurable in Q, is equal almost everywhere in Q<* to a function measurable (33). Finally Fubini's theorem (cf. Chap. Ill, 8) may be stated as follows for the space 0: 160 CHAPTER IV. Derivation of addditive functions. (16.2) Theorem. If f is a non-negative measurable function in the space Q<o, then for any positive integer w, (i) the definite integral jf(, 7?)rfL,,,(/;) exists for erery f e Q m , except at most for those of a set of measure (L m ) zero, (ii) the definite integral //(, rj)dL m (i-) exists for every i\ t Q to , Q m except at most for those of a set of measure (L <0 ) zero, (iii) //(f)*U(0=^ Proof. We begin by verifying this directly when / is the characteristic function of a closed interval, or of a half open interval, and then successively when / is the characteristic function of an open set, of a set (,>)> of a set of measure zero, and finally of any measurable set. It follows at once that the theorem is valid in the case where / is a simple function, and then, by passage to the limit, in the general case where / is any non-negative measurable function. The line of argument that we have sketched, does not differ substantially in any way from the proof of Fubini's theorem for Euclidean spaces, and is even in a sense simpler than the latter, since in proving Theorem 8.1 of Chap. Ill we had to allow for the possibility of there being hyperplanes of discontinuity of the functions U and V. In the space (?,,, there is, however, as shown by B. Jess en [2, p. 273], another theorem of the Fubini type, whose proof requires new methods. This theorem allows integration over the space (/,,, to be, so to speak, reduced to integrations over the cubes (?, in Euclidean spaces, whereas each of the three members of the relation (iii) of Theorem J6.2 contains an integration extended over the space (/,.,. * (16.3) Jessen's theorem. If f is a non-negative measurable function in the space Q^ the integral (16.4) U0=/(f, C)dL ;|| (f ), where m=l, 2, ... , exists, and we have (16.5) for almost all in Q M . [|16J Torus space. 161 Proof. Let us first remark that if Q is a set of measure zero in <2o>, it follows from Theorem 16.2, applied to the characteristic function of Q, that for any m whatever, the set E[(f, rj)Q; SeQm] is of measure (L m ) zero for almost all r\ of Q w . Hence (with the not- ation adopted p. 157) we also have L m {E[(f,C^)eQ]}==0 for almost all C of <2<o. It follows that if g and h are two non-negative measurable functions which are almost everywhere equal in <? w , the integrals /?(f>Q dL m() aixd /MQ*L m (f) are equal for almost all the f Q m <i m of Quj whatever m may be. We may therefore, without loss of gen- erality, assume in the proof of Theorem 16.3 that tue given func- tion / is measurable (33); for any measurable function is almost everywhere equal to a function measurable (93). The integral in the formula (16.4) then clearly exists for every , and moreover it follows directly from this formula that the function /m(C) is cylindrical of order m. The upper and lower limits of the sequence {/ m (C)} are thus cylindrical of every finite order and by Theorem 16.1 we may write almost everywhere in Q^ lim inf / OT (C) =A and lirn sup / m (C) = B m m where A and B are constants. It remains to be proved that A=M=B, where M denotes the integral on the right-hand side of (16.5). We shall prove in the first place that A^M. For this purpose, let A' be any number exceeding A (if A = -f-oo our assertion is obvious), and write (16.6) P*=E[/,(CX-A'], S=P* and S = lim 8 7IJ The set S coincides, except for a set of measure zero, with the whole space Q w . Keeping an index m fixed, let us evaluate the integral of f m overS m . Writing B m =P in ,JB in -i=Pi n -rCP in ,.. M U 1 =P 1 -CP a -...-OP m> we have (16.7) 8 m =ZjR* and (16.8) RrRj=0 whenever On the other hand, since every function /* is cylindrical of order fc, so are the sets PA and CP* and therefore the sets R k for fc=l, 2, ... , m. We may thus write (cf. above p. 159) Rk=QkXRk where jR*C <? According to (16.6), we have fk(t)^A f for every CcR k (2P k where fe=l, 2, ... , m, or, what amounts to the same by formula (16.4), 162 CHAPTER IV. Derivation of additive functions. J j(,rj)dLk(g)^A' for every rjeR/t. Therefore, on account of * Theorem 16.2, we obtain for fc=l, 2, ... , m, * 2*0* whence it follows by (16.7) and(16.8) that Making w->oo, we obtain in the limit Jf=|/(?)dL w (f)^-A' and 80 By symmetry M^B and, since it is clear that A^B, this requires A = MB and completes the proof. CHAPTEE V. Area of a surface % F(x, y). 1. Preliminary remarks. We saw (cf. Chap. IV, 8) that the Lebesgue theory enables us to solve completely the elementary problems concerning the length of a curved line and the expression of this length by an integral. However, similar problems concerning curved surfaces involve difficulties of a much more serious kind. Certain classical treatises on the differential and integral calculus, even in the second half of the XlX-th century, contain an inaccurate definition of the area of a surface. By analogy with the definition of length of a curve, the authors attempted to define the area of a surface as the limit of the areas of polyhedra inscribed in the surface and tending to it. H. A. Schwarz [I, p. 309] (cf. also M. Fr^chet [3]) was the first to remark that such a limit may not exist and that it is possible to choose a sequence of inscribed polyhedra whose areas tend to any number not less than the actual area of the surface. About the same time Peano and Iler- mite subjected the old definition to similar criticisms and proposed new definitions based on quite different ideas. It was H. Lebesgue who first returned in his Thesis [1] to the old method, in a modified form that may be roughly described as follows: the area of a surface is the lower limit of the areas of polyhedra tending uniformly to the surface in question (without, however, being necessarily in- scribed in the latter). Nevertheless, in the more general case in which the surface is given parametrically, this definition requires various additional notions and considerations (cf. T. Bad 6 [I; 1; 1]) and the results obtained are far from being as complete as those available for curves. The difficulties that arise belong to Geometry and 164 CHAPTER V. Area of a surface z=F(x,y). Topology rather than to the Theory of functions of a real vari- able. (For the special case in which the functions x=X(u, t?), y = Y(u, v) and z=Z(u, v) which define the surface parametrically fulfil the Lipschitz condition vide T. Bad 6 [4] and H. Badema- cher [4]). We shall therefore restrict ourselves to the case of continuous surfaces of the form zF(x, y). The most elegant and the most complete results concerning these surfaces are due to L. Tonelli [5; 6; 7]; they will be given in 8 and are the principal object of this chapter. Tonelli 'ft theory is based on the definition of area proposed by Lebesgue. As regards the modern work on area of surfaces based on other definitions, we should mention: W. H. Young [4], J. C. Burkill [3], S. Banach [5], A. Kol- mogoroff [3] and J. Schauder [1]. T. Eado [1, pp. 154 169; 2] has developed further the methods of Tonelli by means of older ideas due to de Geocze and with the help of certain functional introduced by the latter. The principal result of Eado (vide Theorem 7.3), applications of which will be discussed below, enables us to define the area of a surface as the limit of certain simple expressions, whereas the Lebesgue definition only enables us to obtain it as a lower limit. An- other expression is due to L. C. Young (vide below 8) and constitutes a direct generalization of the classical formula for the area of a surface. Except where the contrary is expressly stated, the reasoning of this chapter will be formulated for functions of two real variables. The extension to spaces of any number of dimensions offers no difficulty. $ 2. Area of a surface. By a continuous surface on a plane interval I , we shall mean any equation of the form z=F(x, y), where F is a continuous function on 7 . A continuous surface z=P(x, y) on an interval J is termed polyhedron if there exists a decomposition of I into a finite number of non-overlapping triangles T 19 T 2 , ..., T n such that the function P is linear on each of these triangles, i. e. such that -P(#> y)=<*ix+biy+Ci for (x, y) e T/, where t=O, 2, ..., n and a/, ft/, o t are constant coefficients. We shall call, respectively, faces and ver- tices of the polyhedron z=P(x, y), the parts and the points of the [f 3] The Burkill integral. 155 graph (cf. Chap. Ill, 10) of the function P, which correspond to the triangles T/, and to the vertices of the T,. The sum of the areas of the faces in the sense of elementary Geometry, i.e. the number will be called elementary area of the polyhedron z=P(x, y] on J and denoted by S (P; J ). Given any continuous surface z=F(x, y) on an interval / , we shall term its area on I , and denote by S(^;/ ), the lower limit of the elementary areas of polyhedra tending uniformly to this sur- face, i.e. the lower bound of all the numbers s for each of which there exists, given any >0, a polyhedron z~P(x y y) on 7 such that 1 \P(x, y) F(x, y)\<e at every point (a?, y) e 7 and 2 We might verify nere that for polyhedra the elementary area agrees with the area according to the general definition just given. As, however, this is an easy consequence of the theorems given further on (vide p. 181), a special proof is unnecessary at this point. It should be remarked that, in accordance with the definition adopted, the area of a surface may be either finite or infinite. The following theorem is an immediate consequence of the definition. (2.1) Theorem. For any sequence of continuous functions {F n } which converges uniformly on an interval Z to a function P, we have 3. The Burkill Integral. Instead of treating the theory of area of surfaces by itself, it is more convenient to associate it with certain differential properties of functions of an interval. How- ever, the functions of an interval occurring in the theory of area are not in general additive, and in consequence we shall have to complete in some minor points the theory of functions of an interval, developed in the two preceding chapters. We shall begin with some subsidiary definitions. To simplify the wording we shall understand in the sequel by subdivision of a figure R Q any finite system of non-overlapping intervals I 19 / 2 , ... , //i such that Ro=lk- Given any function of an interval U and given a finite system of intervals 3={//r), we shall write, for brevity, 17(3) in place of U(Ik)- In particular therefore, L(3) will denote the * sum of the areas of the intervals belonging to the system 3- 166 CHAPTER V. Area of a surface z=F(x,y). We call upper and lower integral in the sense of Burkill of a Junction of an interval U(I) over a figure R OJ and we denote by JU and JU respectively, the upper and the lower limit of /?c S"u the numbers 7(3) for arbitrary subdivisions 3 of JK , whose charac- teristic numbers 4(3) tend to zero (cf. Chap. II, p. 40). When these integrals arc equal, their common value is called the Burkill definite integral (or simply the integral) of the function U over jR and is denoted by fu. If this integral exists and is finite, the func- *0 tion U is said to be integrable on R Q (in the sense of Burkill). If the function V is integrable on every figure R (in the whole plane or in a figure R ) its integral / U considered as a function of the figure R R is called indefinite integral of U (in the whole plane or on 7? ). (3.1) Theorem 1 If U is a function of an interval and R ly J? 2 are non-overlapping figures, we have (3.2) fU2/U+-fu and provided that both integrals of U over R^R^ are finite. 2 Any function of an interval U which is integrable on a figure R , is equally so on every figure R(^R a nd its indefinite integral on R is an additive function of a figure. Proof, Part 1 of the theorem is a direct consequence of the definition of the Burkill integrals, and part 2 follows at once from the formulae (3.2) when we subtract the second of these formulae from the first. If U is a function of an interval on a figure R, we shall call variation of V on R at a set D the upper limit of 1 17(3) I as 4(3) ->0, where 3 denotes any finite system of non-overlapping intervals contained in R and possessing common points with D. The follow- ing analogue of Theorem 4.1, Chap. Ill, may be noted. (3.3) Theorem. Given on a figure R Q a function of an interval U such that /|i7|< + oo, there can be at most an enumerable infinity of x* straight lines D, which are parallel to the coordinate axes and at which the variation of U on RQ is not zero. [f 3] The Burkill integral. 167 In fact, the number of straight lines which are parallel to the axis of x or of y, and at which the variation of U on R exceeds a positive number e, cannot be greater than 2e~ 1 ' j\U\< + oo. RO (3.4) Lemma. Given a junction of an interval U integrable on a figure R , there exists, for each e > 0, an q > such that for every system ^={1^ I 2 , ... , I p } of non -overlapping intervals situated in R Q9 the inequality zl(3)<?? implies the inequality (3.5) *=, Proof. Let ??>0 be a number such that, for every subdivision I of J2 , J($)<q implies \U(1) fu\< e/2, and let 3=!/i, I - , I P } * be any finite system of non -overlapping intervals situated in JK , such that J(3) < i?. Let R l =R Q Q^I k . By Theorem 3.1, the function k-=\ U is integrable on JB 1 . It follows that there exists a subdivision 3i of R l such that (3.6) Now 3 + 3i clearly constitutes a subdivision of R such that i)<*?. We therefore have |*7(3 + 3i) / '?7|<c/2, and we need only subtract the second of the relations (3.6) from it to obtain (3.5). If jR is a fixed figure, then to any ??>0 there corresponds a positive integer p such that every interval /C^o ma y be sub- divided in p subintervals of diameter less than rj. Hence applying Lemma 3.4, we obtain at once the following (3.7) Theorem. If a function of an interval U(I) which is in- tegrable on a figure 12 , is continuous, then the same is true of its in- definite integral B(R)=fu. K (3.8) Theorem. If U is a function of an interval which is in- tegrable on a figure R Q and if B is its indefinite integral, then S(x J y)=^U(x,y) and B(x,y)=U(x,y) at almost all points (x,y)eR 9 . In particular therefore, if one of the functions U and B is almost everywhere derivable in R , the same is true of the other and the deriv- atives of U and of B are almost everywhere equal. 168 CHAPTER V. Area of a surface z=F(x t y). Proof. Suppose that the set of the points (#, y) at which U(i y) >-(#, y) has positive measure. We could then determine a set E(^R of positive outer measure and a number a> 0, such that U(Xj y) J3(#, y)> a at each point (x,y) of E. Therefore, on account of Vitali's Covering Theorem (Chap. IV, Theorem 3.1), we could determine in JE , for any r\ > 0, a finite system of non- overlapping intervals 3 =!/*}*-! ,2,..., n such that J(3)<^, L(3)>'^;/2, and U(I k ) B(I k )>a-\Ik\ for i=l, 2, ... , n. Now it follows from the last two relations that 17(3) #(3)> a-|JS?|/2, which contradicts Lemma 3.4. Hence, U(x, y)*^B(x, y) almost everywhere in R Q . In the same way we prove that the opposite inequality holds also almost everywhere in R , and this completes the proof. (3.9) Theorem. Suppose that U is a continuous function of an interval on a figure R and that (i) /|E7l<-foo and (ii) for every interval IC^o an ^ every subdivision 6 of I. Then the func- tion U is integrable on R$. Proof. Given a number 6>0, let $= {7/}^i,2,. ,/> be a sub- division of RQ such that (3.10) U(Z)>fU e. t*n Let us denote by Di, /> 2 , ... , D r the sides of the intervals (1) which do not belong to the boundary of 7? . By Theorem 3.3 it may be assumed, in view of the continuity of the function U and of condition (ii), that the variation of U on R Q vanishes at each side />/ It follows that there exists an rj > such that, given any finite system S of non-overlapping intervals situated in # and having points in common with the sides Z),, the inequality J(S)<ij im- plies |Z7(6)|<. We can clearly assume that rj does not exceed the length of any side of the intervals (5). This being so, consider an arbitrary subdivision 3 (A> ^2* ? ^) of # such that 4(3)< rj. By numbering the intervals of 3 suitably, we may evidently suppose thai; 7 1? / 2 , ... ,I V are those having points in common with the sides /) while the remaining intervals of 3 (if any) have none. Finally, let us agree to write U( J/O W = when J/QI* = 0. Then \u(I k )\<e and \ V(Ji<^l*)\<** so [ 4] Bounded variation for functions of two variables. 169 that, by (3.10) and by condition (ii) of the theorem, we have rU 8<U(Z)^U(3)2U(I*)+S SU(J t QI k )<V(3) + 2e. It jf o *=i /=i A=I follows that fu^fu+3e, and so, that fu = fu. RQ RQ R R In connection with this , vide J. C. Burkill [2; 3; 4], R. C. Young [2] and F. Riesz [6; 7]. 4* Bounded variation and absolute continuity for functions of two variables. Given a function F(x, y) contin- uous on an interval I=[a^ ft t ; a 2 , ft 2 ], let us denote for any value x subject to a^^x^bi, by W 1 (.F; x; 2 , b 2 ) the absolute variation of the function F(x, y) with respect to the variable y on the interval [a 2 , b 2 ]j and for any value y subject to # 2 ^y^fc 2 , by ^2(^5 #5 a \i M that of the function F(x,y) with respect to x on [a ly fr x ]. Denoting by Jj and J 2 respectively the linear intervals [a 1? fc t ] and [# 2 , ft 2 ] we shall also write W^jPja?; J 2 ) for W^J?;^; a 2 , ft 2 ) and W^F;*/;^) for W 2 (F; y; a,, ft^- By continuity of the function JP 1 , the non-negative expressions W 1 (J T ; ^; 7 2 ) and W^Fji/jJj) are, as is easily seen, lower semi- continuous functions of the variables x and y respectively. When 6, 6 2 the integrals JW^F^x^J^dx and jW 2 (F}y;J l )dy are both finite, a\ <i> the function jP 7 is said to be of bounded variation on / in the Tonelli sense. It follows at onfce that any function of bounded vari- ation of two variables x, y is of bounded variation with respect to x for almost every value of y and with respect to y for almost every value of x. A continuous function F(x^y) will be termed absolutely con- tinuous on an interval /=[#i, b^a^b^\ in the Tonelli sense, if it is ol bounded variation on / and moreover, absolutely continuous with respect to x for almost every value of y in [0 2 , 6 2 ], and absolute- ly continuous with respect to y for almost every value of x in [o^ 6J. We say that a function F(x, y) fulfils the Lipschitz condition on J, if there exists a finite constant N such that \F(x',y')F(x",y")\^ <:.#(#' x"\+ y' y"\) for every pair of points (x', y') and (x",y") of I. Any function which fulfils the Lipschitz condition on an interval / is evidently absolutely continuous on I. In particular 170 CHAPTER V. Area of a surface *=F(x,y). polyhedra and also functions of two variables with continuous partial derivatives, are always absolutely continuous functions. A function F which is continuous and of bounded variation [absolutely continuous, or subject to the Lipschitz condition] on an interval /o=[<h>&i; #27*2] can easily be continued, even so as to be periodic, over the whole plane in such a manner as to remain continuous and of bounded variation [absolutely continuous, or subject to the Lipschitz condition] on every interval. In fact, de- noting by I I one of the intervals congruent to / with a common side parallel to the axis of #, let us continue the function F from the interval / on to the interval I I by symmetry relative to the common side of these intervals. Let us further continue similarly the function F from the interval / +A on to an interval Z 2 con- gruent to IQ+II which has with the latter a common side parallel to the axis of y. The function F is then defined on the interval 10+1!+ / 2 whose sides are respectively of lengths 2-(& 1 a t ) and 2-(6 2 a 2 ). Writing w = 2-(6 1 %) and v = 2-(b 2 a 2 ), and continuing the function F from the interval /o+j^+Ig on to the rest of the plane by the periodicity condition F(x+ u, y)=F(x y y+v)F(x y t/), we see easily that the continuation obtained for the function F has the properties required. Besides the definition of Tonelli several other definitions have been given of conditions under which a function of two variables is said to be of bounded variation. For a discussion of these definitions see C. R. Adams and J. A. C larks on [1;2]. Throughout this chapter, use is made of Tonelli's concept only. We shall subsequently make use of the following theorem concerning the partial derivates of any continuous function: (4.1) Theorem. Given a continuous function F(x, y), its partial Dini derivates j F^jF^F^F^ and Fy,F^,F^,F^, are func- tions measurable (33). Proof. It will suffice to prove this for any one of these deriv- ates, say F*. Given an arbitrary real number a, consider the set and denote by E n the set of all the points (#, y) such that for every h the inequality 0<A^l/n implies [F(x+h,y) F(x, y)]lh^a 1/n. [ 5] The expressions of de Geocze. 171 We find that E=E n and, since by continuity of the function F n each of the sets E n is closed, E is a set ($<,), so that the derivate F+ is a function measurable Theorem 4.1 may be compared with Theorems 4.2 and 4.3 of Chap. IV concerning measurability of the derivates of functions of one real variable. Never- theless it is to be remarked that contrary to what occurs for functions of one variable, the partial Dini derivates of a function measurable (B) need not in general be measurable (93), although they are still measurable (0) (the proof of this requires, however, the theory of analytic sets; vide F. Hausdorff [II, p. 274], M. Neubauer [1] and A. E. Currier [1]). On the other hand, a function of two variables may be measurable (2) without its partial Dini derivates being so. $ 5. The expressions of de Ge5cze. We shall make correspond to any function F(x, y) which is continuous on an interval !=[*!, 6j; a 2t ft 2 ], the following expressions introduced by Z. de Geocze [1] into the theory of areas of surfaces: ( F; While studying the fundamental properties of these expressions, we shall often find the following two inequalities useful: (5.D for any three sequences {#/}, {y/} and {z/} of real numbers; (5.2) [(/**)(/y*)V(/^ E E E R for any measurable set E in a space R m and any three non-negative functions x(t), y(t) and z(t), measurable on E. The inequality (5.1) is easily deduced by induction from the case n = 2 which can be verified directly. The inequality (5.2), in the special case in which the functions x(t), y(t), z(t) are simple, is an obvious consequence of (5.1); and we pass at once to the general case with the help of Theorems 7.4 and 12.6 of Chapter I. 172 CHAPTER V. Area of a surface z=F(x, y). (5.3) Theorem. The expressions of de Geocze (7 1 (I)=G 1 (J 7 ; I), {? 2 (Z) G 2 (-F; I) and 0(I) = G(F; I), associated with a continuous function F(x, y), are continuous junctions of the interval I and their integrals over any interval exist (finite or infinite)] these integrals over any interval / =[ a i> *i> a zj M fulfil the following relations: (5.4) O^WjKFiX-.^b^dx and fo (5.5) fly / /O /O (5.6) Proof. Given an arbitrary e>0, let 77 < be a positive number such that, for any pair of the points (#, t/j) and (#, y 2 ) in I , (5.7) |# 2 yj(<ri implies \F(x, y 2 ) F(x, y l )\<e. Let M denote the upper bound of F(x,y) on J and consider in 7 an interval /= [a l9 fa a 2 , j9 2 ] such that |I|<7y 2 . We then have either & a 1 <?; or /S 2 a 2 <rj. In the former case, we find 0^1)^(0! a l )-2M<2Mri^.2Me, and in the latter we derive from (5.7), G 1 (I)^(0 l a 1 )-g^(6 1 a^-e, so that in both cases 1 (/)^(23/ + 6 1 a^-e. The function O^I) is therefore continuous. The same is of course true of 2 (I) and the continuity of these two functions at once implies that of G(I). This being so, we shall show that the functions O l and (? 2 are integrable and, at the same time, we shall deduce the formulae (5.4). Let {3n} be a sequence of subdivisions of I such that lim A (3J = and \imG l (3 n )= I G^ and denote, for any positive /i n fo integer n and any fe[0i&i]> by F n (f) the sum of the absolute increments of the function F(, y) on the linear intervals cut off on the line x= by the rectangles of the subdivision 3/i We then have, on the one hand, *i (5.8) 0i(3J=/V n (f )dS for n=l, 2, ... , , and on the other hand, on account of continuity of the function F, lim F n (f) = W 1 (J? T ; f; a 2 , 6 a ) for any f [a 1? 6J. Therefore, in virtue of [f 6] The expressions of de Geocze. 173 Fatou's Lemma (Chap. I, Theorem 12.10) and by (5.8), we obtain 6, 6, . But since O^^w^^a^b^df for every subdivision 3 of / , we have also G^w^F; f ; a 2 , fc 2 /(> l Therefore the function G l has a unique integral over I and this integral fulfils the first of the relations (5.4). The existence of the integral JG 2 and the validity of the second of these relations are /o deduced by symmetry. Let us pass on now to the function 0. We first remark that the integral fo clearly exists in the case in which one at least of /o the integrals fG l and f(} 2 is infinite, and is then also infinite fo fo on account of the relations (5.9) #i(/X G(I) and G 2 (I)^G(I) for any interval I. In the remaining case, the two integrals in question being finite, the evident inequality ^(/X^IJ + ^/J + III yields (5.10) /o /o /o and on the other hand, for every subdivision 3 of any interval I the equally obvious relations (5.11) 0i(/)<0i(3) and # a (Z)? 2 (3) lead, in view of the inequality (5.1), to (5.12) Now, continuity of the function G being already established, the formulae (5.10) and (5.12) imply, by Theorem 3.9, that this function has over I a unique integral. To complete the proof we need only remark that the formulae (5.11) and (5.12) imply at once the formulae (5.6) and finally that formula (5.6) follows directly from (5.9) and (5.10). As a corollary of Theorem 5.3, and more particularly as a con- sequence of the formulae (5.4) and (5.6), we have: 174 CHAPTER V. Area of a surface z (5.13) Theorem. In order that the function of an interval G(I)= associated with a continuous junction F(x, y), be integrable on an in- terval / (i. e. in order that J (?<+)> it is necessary and sufficient /o that the function F(x,y) be of bounded variation on I . 6. Integrals of the expressions of de Geftcze. Given a continuous function F(x, y), we shall denote, for any interval I 0f by H^JPj/o), H 2 (F;/ ) and H(JF; 7 ) respectively, the integrals ol the functions of an interval <? 1 (/)=G 1 (J 7 ; I), G 2 (I)=Gt(F; I) and #(I)=G(jF; /) over the interval 7 . All these integrals exist on account of Theorem 5.3 and their importance in the theory of area of surfaces is due to the fact that, as will be shown in the next , the number H(jP; / ) coincides with the area of the surface z~F(x, y) on Z . (6.1) Theorem. For any function F(x, y) which is continuous and of bounded variation, the expressions JSf^IJ^H^jF;/), H 2 (I)=H 2 (F;I) and H(I)=H(F;I) are additive, continuous, and non-negative func- tions of the interval I, and we have at almost all points (x,y) of the plane H'(x, y) = Proof. Additivity and continuity of the functions in question follow at once from Theorems 3.1 and 3.7 on account of Theorem 5.3. We have therefore only to establish the relations (6.2). Now, for any interval I=[o 1 , b^ a 2 , fr 2 ] we have according to Theorem 5.3 and Theorem 7.4 of Chap. IV, the following relation (in which the transformation is effected in accordance with Pubini's Theorem 8.1, Chap. Ill, rendered applicable to the partial derivates of the func- tion F(x,y) by Theorem 4.1): HAD^fWAF-, *; a 2 , b,)te^f[f\F(x, y)\ dy] dx=ff\F y (x, y)\ etedy; Oi a, a, I whence (6.3) H\(XJ y)^*\F' y (x, y)\ for almost every point (x, y). Let us now denote by { J n } the sequence of the linear intervals with rational extremities. In view of Theorem 7.4, Chap. IV, we [6] Integrals of the expressions of de Geocze. 175 have for n=l 9 2, ... and for every linear interval J, ; a; J a ) dx^H^J xJ n ) ^ f fs\(x, y)dxdy^ f [/>,(#, y) dy\ dx, * j J and consequently, for each positive integer n, the inequality W l (F;x;J n )'^JH'i(x,y)dy holds at every point x, except at most those of a set E n of linear measure zero. Therefore, writing E=< n we obtain the inequality W^ F\ x\ J) > /#i(#, y) dy, whenever J has rational extremities and x lies outside the set E of linear measure zero. If we now regard the two sides of this inequality, for a given value of x outside the set J3, as functions of the linear interval J, we obtain by derivation with respect to this interval (on account of Theorem 7.9, Chap. IV) for almost all y, the inequality (6.4) \F y (x,y)\^H\(x,y). Therefore, since the derivatives H\(x^y) and F' y (x, y) are meas- urable (cf. Theorem 4.1), it follows from Fubini's theorem (in the form (8.6), Chap. Ill) that the set of the points (#, y) at which the relation (6.4) is not fulfilled, is of plane measure zero. By (6.3) we therefore have almost everywhere the first of the relations (6.2). The proof of the second relation now follows by symmetry, and that of the third from the remark that if we write fl ( 1 (/) = G 1 (JP; I), 2 (Z) = G 2 (JF;/) and G(I) = G(F; I), we have by Theorem 3.8, at almost every point (x, y) of the plane. This completes the proof. (6.5) Theorem. In order that the function of an interval H(I)=H(F;I), corresponding to a continuous function F(x,y) of bounded variation on an interval I =[o 1 , b^ a 2 , 6 2 ], be absolutely continuous on this interval, it is necessary and sufficient that the function F(x,y) itself be absolutely continuous; and when this is the case, we have (6.6) JET (J ) =ff{lK(x, y)] 1 + [F y (x, y)] 2 + l} 1/f dxdy. f. 176 CHAPTER V. Area of a surface z=F(x, y). Proof. By Theorem 5.3, absolute continuity of the function H(I) is equivalent to absolute continuity of the functions H^I) and H 2 (I) together. Therefore if the function H is absolutely continuous on I , we have, by Theorem 6.1, for any interval J|=[%, ; a a , ft a ], where the relation i &, ; a a , 6 2 ) dx^H^I^^fflF^x, y)\ dx dy=f \J \F' y (x, y)\ dy] (to, and, taking the derivative with respect to I, we obtain for almost every value of #, (6.7) W^F; X-, a b 2 )=f\F y (x, y)\ dy. Now, for any given value of x (for which F(x,rj) is of bounded variation in rj) the difference W^jF; x\ a 2 , rj) J\F y (x, y)\ dy is a non- negative and non-decreasing function of the variable r\ (cf. Theo- rem 7.4, Chap. IV). It therefore follows from (6.7) that we have for almost every value of x, and for any 77 e [a 2 , ft a ]> i. e. that the function W 1 (J< T ; a?; a 2 , ??), and consequently also F(x, y), is absolutely continuous with respect to r\ on [a 2 , 6 2 ] for almost every value of x. By the symmetry of the variables, we conclude also that the function F(g, y) is at the same time absolutely contin- uous with respect to f on [oj, ftj] for almost every value of y. The function F, which is by hypothesis of bounded variation on I , is therefore absolutely continuous in the Tonelli sense on this interval. Conversely, if the function F is absolutely continuous on I , we have by Theorems 7.8 and 7.9, Chap. IV, for every subinterval I=[ ai , ft; a 2 , j8 2 ] of / , the relations: ^I^fw^F; X-, ct 2 , p 2 ) dx=ff\F' y (x, y)\dxdy, ^f f\F' x (x, y)\dxdy, so that the two functions of an interval H^ and T 2 , and therefore also a, are absolutely continuous. [7] Rad&'a Theorem. 177 Finally, since the function H is absolutely continuous, the for- mula (6.6) is a direct consequence of the third of the relations (6.2), the latter being valid almost everywhere on account of Theorem 6.1. Up to the present we have regarded the expression H(JP;Z) as a function of an interval I. If we treat this expression as a func- tional depending on the function jF, we obtain the following theorem, whose geometrical interpretation will appear in 8 when the theorem appears to be a generalization of Theorem 2.1. (6.8) Theorem. Given any sequence of continuous functions {F n } which converges to a continuous function F, we have for every interval I (6.9) lim inf H(^ n ; I) > H(,F; I). /i Proof. Denoting by 3 P the subdivision of I into p 2 equal intervals, similar to /, we have by Theorem 5.3 for any pair of in- tegers p and n, H(J? n ; I)^G(F n ; 3/>), and by Fatou's Lemma (Chap. I, Theorem 12.10), for every integer p, lim inf G(F n -, 3 P )>G(J^; 3,)- We therefore have lim inf H^; I)^G(J^; 3 P ), and this leads to n (6.9) when p->oo. $ 7. Radd's Theorem. Before passing to the proof of the result of Ead6, according to which the area of any surface z=F(x, y) on an interval I is equal to H(^;/), we shall prove the following (7.1) Theorem. If a continuous function F(x,y) has on an interval J = [a 1? b^ a 2 , 6 2 ] continuous partial derivatives, there exists a sequence of polyhedra {z=P n (x, y)} inscribed in the surf ace zF(x,y), such that the sequence converges uniformly to this surface and such that (7.2) lim S (P n ; 7 ) = ' Proof. Let 3* = {1.1* In& , !,'} denote the subdivision of I into n 2 equal intervals similar to I , and (x njj y n , ,), where t =1, 2, ... , n 2 , the lower left-hand vertex of !,/. Let us divide any interval /,/ into two right-angled triangles T' n j and T^/ by a diagonal, in such a way that the vertex (#,/, y n , t ) is that of the right angle of T Htt . Consider for any n the polyhedron zP n (x 9 y) inscribed in the surface z=F(x, y) and corresponding to the net formed on 7 by the 2n 2 triangles T' nii and T'nj where t=l, 2, ... , n 2 . 178 CHAPTER V. Area of a surface =jP(x, y) For brevity let hn^^a^jn and fc n =(& 2 a 2 )/n; and let fA n denote the upper bound of the differences \F' x (x" y y") F' x (x' 9 y')\ and \F' y (x", y")F' g (x', y')\ for all points (x', y') and (#", y") of I such that \x" x'\^Kn and \y" y'l^foi. Now if 8 n ,t and s'n t t denote respectively the elementary areas of the faces of the polyhedron z=P n (x, y) which correspond to the triangles T t / and T^',/, we notice at once that the areas of the pro- jections of the former of these faces on the planes xz and yz are respectively equal to , y n ,i)\=khnlcn'\Fy(x nii , y nj )\ and n , y n ,i)F(x n , h y n ,i)\ = kh n kn'\F' x (x n ,t, y n j)\ where #, i^x n j^x n j+ h n and y n , / < y' n , / ^ y n , i We therefore have 8 n ,i=l{[F*(x n ,,,y ,,)]* + [^( and so, by the inequality (5.1), p. 171, n* n* 2*'.^2m*n t l,yn.itf^ t=\ i^=\ Since the partial derivatives F' x and F' y are by hypothesis continuous, it follows by making n->oo that lim s' n ,i= W*> y)f+[^(^ y)f+l} l *dxdy, t=l /o and the same limit is clearly obtained for the sum of the s'nj. By addition, together with an appeal to Theorem 6.5, we now derive the formula (7.2) and this completes the proof. In what follows we shall apply the method of mean value integrals. Given in the plane a summable function F(x,y), the 1 n In sequence of functions F n (x, y) n* I JF(x + u, y+v)dudv where o o n=l,2, ..., will be called sequence of mean value integrals of the function F(x, y). It is clear that if the function F is continuous, (i) the sequence of its mean value integrals (F n (x,y)} converges to F(x, y) at every point (a?, y) of the plane, and uniformly on any interval, and (ii) the partial derivatives 3F n j3x and 3F n /3y exist everywhere and are [7J Rad&'a Theorem. 179 continuous. In fact, at any point (#, y) a direct calculation gives l/n dF n (x,y)l9x=n z f[F(x+l/n, y + v)F(x, y+vftdv and l/ii , y+i/ n )-F(x+u, y)]du. It was T. Rad6 [2] who first applied in the theory of area of surface* the method of mean value integrals. The role of these mean values is due to the fact that in the case in which the given function F is continuous, the sequence of areas of the surfaces z~F n (x,y) on any interval tends to the area of the surface z=F(x t y) (vide, below, Theorem 7.3). In the definition given above, the functions F n are defined at each point (x, y) as "mean values** of the function F over squares of which (x, y) is a vertex; it goes without saying that we could also make use of mean values taken over squares, or circles, having (x, y) as their centres. These mean values over circles are used for instance in potential theory (cf. F. Riesz [4] and G. C. Evans [1]). (7.3) Radb'a Theorem. If F(x, y) is a continuous function and (F n (x,y)} is the sequence of mean value integrals of F(x,y), then (7.4) H(P; J ) = 8(J T ; J )=lim 8(^5 J ) n for every interval J . Proof. Let {z=P n (x, y)} be a sequence of polyhedra converging uniformly to the surface z=F(x,y), such that (7.5) lim8 (P ll ;I ) = S(jP;I ). n Since the functions P n (x, y) are absolutely continuous, it follows from Theorem 6.5 (cf. also 2, p. 165) that S (P,,; I Q ) = H(P n ; I ) for every n. Consequently, since the sequences of functions {F n } and {P n } converge uniformly to the function F, it follows by using successively Theorem 2.1, the formula (7.5) and Theorem 6.8, that (7.6) liminf8(^;/ )>S(J? 7 ;Io)=limH(P /J ;I )^H(^;/ ). n n Now if the function F is not of bounded variation on Z , it follows from Theorem 5.13 that H(jP; I )=-foo and consequently the formula (7.4) follows at once from (7.6). We may therefore as- sume that the function F is of bounded variation on I , and further (cf. 4, p. 170) that F is continuous and of bounded variation on each interval of the plane. 180 CHAPTER V. Area of a surface z=*F(x, y). Let us agree to denote, for any set Bin the plane, by E (u ' o) the parallel translation of E by the vector (u, v) (cf. Chap. Ill, 11); similarly, for a family of sets in the plane, (g (UfU) will denote the family of all the sets obtained from sets (() by subjecting them to this translation. For any subinterval I=[a 1 , brf a 2 ? &] we obtain i ; I)=f\F H (x, b 2 )-F n (x, a, 00 1 /i 1 n and a similar formula for G 2 . Hence by the inequality (5.2), p. 171, 1/1 1 71 f /{[G^F; I (u ' u) tf 1 n = n*f fQ(F',l <u ' } )dudv. 1 n 1/n Denoting by 3/> the subdivision of / into p 2 equal intervals similar to I , we obtain therefore, for every p, 1 n 1 n G^SpXn 8 // 6 6 and since by Lemma 3.4, G(F;3{." >0) ) tends to uniformly in u and v, we obtain in the limit (7.7) as p- Finally the areas of the figures Io"' P) -3/o and I 3/o"' u) tend to with w and v and each of these figures is a sum of two intervals. Hence since the expression H(^;7) is by Theorem 6.1 a contin- uous function of the interval I, we have lim H(l fT ;Io lfW) ) = H(J 7 ;Io). u->0, -M) On the other hand, since the functions F n (x, y) have continuous partial derivatives, we have, by Theorem 7.1, S(F n ; Io)^H(^;Io) for each n. Therefore making n->oo in (7.7), we find lim sup S(^/,; /o)^ n ^H(jP;/ ), which in conjunction with (7.6) gives the required relation (7.4). [8] Tonelli's Theorem. 181 8. Tonelli's Theorem. The theorem of Kado just established, enables us to replace in all the theorems of this chapter the ex- pression H(JF;Z) by the surface area S(F;I). Thus for instance, Theorem 6.5 (formula (6.6)) expresses the fact that the elementary area of a polyhedron coincides with its area according to the general definition of area of a surface. Theorem 6.8 contains a generalization of Theorem 2.1; it enables us to replace in its statement uniform convergence by ordinary convergence: we thus obtain a theorem similar to Lemma of Fatou (Chap. I, Theorem 12.10). It follows that the uniform convergence of the inscribed polygons, required in the definition of area, may be replaced by the ordinary convergence, so that the area of a contin- uous surface zF(x^y) is the lower limit of the areas of polyhedra tending to this surface. Further, by Theorem 7.1, if a function F(x, y) has continuous partial derivatives, there exists a sequence of polyhedra inscribed in the surface z=F(x, y), tending uniformly to the latter and having areas which converge to the area of this surface. (For further generalizations vide S. Kempisty [1]. Cf. also on this subject H. Eademacher [3], W. H. Young [5], M. Fr6chet [2] and T. Eado [5].) Finally, we obtain the following theorem, which sums up the most essential considerations of this chapter: (8.1) TonelWs Theorem,, a) In order that a continuous surface z=F(x,y) have a finite area on an interval I , it is necessary and sufficient that the function F(x,y) be of bounded variation on I . b) When this is the case, we have the expression 8(1) S(F; I) is then an additive continuous function of the interval /C^o an ^ we have for almost every point (x,y)eI c) In order that we should have r r\ I ^F\^ / 5 P \ 2 2 (8.2) it is necessary and sufficient that the function F(x, y) be absolutely continuous on / ; and in order that this be the case it is necessary and sufficient that the area S(F;I) be an absolutely continuous function of the interval 182 CHAPTEK V. Area of a surface z=F(x,y). Proof. The assertion a) follows directly from Theorem 5.13; b) and c) follows from Theorems 6.1 and 6.5 on account of Theo- rem 7.4, Chap. IV. With regard to Theorem 8.1 vide L. Tonelli [5;6;7]. The necessity of con- dition a) was established a little earlier by G. Lampariello [1]. According to Tonelli's theorem, the relation of equality (8.2) can hold for a continuous surface z=F(x,y) only in the case in which the function F is absolutely continuous. Nevertheless, as proved by L. C. Young, this relation will remain valid for arbitrary continuous surfaces, as soon as we replace on the right-hand side the partial derivatives by ratios of finite differences and transpose the passage to the limit outside the integral sign. In fact: (8.3) Theorem. For any continuous surface z=F(x,y) and any interval J we have (8.4) S(^Z )=lim f /-([*(*+.)-*(*,) n^^Qj J [[ a and in order that the function F be of bounded variation on J , it is necessary and sufficient that (8.5) lim sup TT ^ W f f\F(x+a, y+p)F(x, y)\dxdy ,/*-*> \<*\ + \P\J J <+oo. Proof. Let {F n } be the sequence of mean value integrals (cf. 7, p. 178) of the function F. Denote, for brevity, by R(x,y; a,) the expression under the integral sign on the right-hand side of (8.4), and by R n (x, y\ a, /?), for each positive integer n, the expression obtained from R(x,y',a,p) by replacing F by F n . Finally let us write for w=l, 2, ... E n (x, */)=lim R n (x, y; a, P)={[dF n (x, y)ite]*+[9F n (x, y)l3y] 2 +l} 1 ' 2 . , 0-M) In order to establish the identity (8.4), it evidently suffices to show that (8.6) S(JF; J ) < lim inf f f R(x, y; a, p)dxdy ,W J and (8.7) S(^;7 )>limsup f [lt(x,y\ a, 0)dxdy. - J [8] Tonelli's Theorem. 183 For this purpose, let I be any interval in the interior of I . By means of the inequality (5.2), p. 171, we easily find that /Q Q\ J?//T i 1/rt/?\ <T* /W Now let n be a positive integer, sufficiently large in order that (#, y) Ij \u\ < 1/n and |t>|< 1/n should imply (x+u, y+v) I . We then ha,veR(x+u,y+v; a^)dxdy^ li(x,y, a, fi)dxdy and , consequently, by (8.8), J fR n (x,y,a,p)dxdy*^ ffR(x,y, a,ft)dxdy. I /o Making a->0 and /3~>0, we obtain in the limit S(^n;/)^ ^liminf / / R(x, y; a, fi)dxdy. This relation being thus established ,/9-X) ^/ 'o for each interval 1C Jo, we may replace, on its left-hand side, I by Z , and making still n~>oo we obtain the relation (8.6). In order to prove the relation (8.7), let us first observe that the latter is obvious in the case in which S(J? 7 ;I )= + oo. We may therefore assume that the function F(x,y) is of bounded variation on I and moreover (cf. 4, p. 170) of bounded variation on every interval in the plane and periodic with respect to each variable. We can therefore determine an interval J Q =[a l ^ b^ a 2 , 6 2 ] containing I in its interior, such that its sides ^ % and b 2 a 2 are the periods of F(x,y) with respect to x and y respectively. This being so, we find easily, on account of the inequality (5.1), p. 171, that, for any pair of positive integers n and fc, By integrating the two sides of this inequality over J , and taking account of the periodicity of the function F, we obtain ff Jo R tt (x,y, and hence, passing to the limit, making first fc-oo, and then n->oo, we find by Kadd's Theorem 7.3, 184 CHAPTER V. Area of a surface z=F(x, y). fj'R(x,y-,a,p)dxdy^limfj'R n (x,y)dxdy=]imS(F n -,J )=S(F-,J ), /o A> and so Urn sup / lR(x,y; a,fi)dxdy^$(F; <7 ). a,(?-N> ' / J o Now, by the result already established in the inequality (8.6), we have lim inf / / R(x, y; a, ft) dxdy ^ S( F; I) for every interval I. , p-x> J i It follows at once that (8.9) remains valid when we replace the interval J by any subinterval of J , and in particular by the interval I . We thus obtain the relation (8.7). Finally let us remark that on account of the relation (8.4), in order that the area of the surface z=F(x,y) on I be finite, it is necessary and sufficient that lim supyj / / }F(x+a, y)F(x, y)\dxdy <+oo -M) \<*\J J / o and lim sup^- f f\F(x, y+p)F(x, y)\dxdy ?-#> \P\J J <+oo. Now this pair of relations is easily seen to be equivalent to the relation (8.5) which therefore expresses a condition necessary and sufficient in order that the function F should be of bounded variation on I . This completes the proof. A statement analogous to Theorem 8.3 can be made for curves (cf. Chap. IV, 8). If C is a continuous curve defined by the equations xX(t), y=Y(t) t its length on any interval I Q =[a,b] is given by the formula (8.10) S(;/ ) = lii In particular therefore, in order that a continuous functions G(t) be of bounded variation on an interval [a, 6], it is necessary and sufficient that b (8.11) lim sup * / \G(t+h)-G(t)\dt<+oo. h-X) \ n \J [ 8] TonelK's Theorem. 185 This assertion can be proved by the method of mean value integrals in a man- ner quite similar to that we made use of in the theory of areas of surfaces z=F(x, y), but for curves this method can be very much simplified. Let us observe further that the relation (8.11) may be interpreted in a more general sense. In fact, given any summable function G(t), the relation (8.11) is the necessary and sufficient condition in order that the function G be almost everywhere on [a, 6] equal to a function of bounded variation (vide G. H. Hardy and J. E. Little wood [1]; cf. also A. Zygmund [I, p. 106]). Finally the relation (8.10) holds for any rectifiable curve given by the equations x=X(t), t/ = T(/), where the functions X(t) and Y(t) are not necess- arily continuous, provided however that for each t the point (X(t), Y(t)) lies on the segment joining the points (X(t~), Y(t)) and (X(t+) 9 F(f+ )). CHAPTEE VI. Major and minor functions. 1. Introduction. Major and minor functions (defined in 3 of this chapter) were first introduced by Ch. J. de la Valise Poussin in his study of the properties of the Lebesgue integral and those of additive functions of a set. Entirely equivalent notions (of "Ober"- and "Unterfunktionen") were introduced independently by O. Perron [1], who based on them a new definition of integral, which does not require the theory of measure. Although, in its original form, this definition concerned only integration of bounded functions, its extension to unbounded functions was easy and led, as shown by O. Bauer [1], to a process of integration more general than that of Lebesgue. Moreover, as we shall see in 6, the Perron integral may be regarded as a synthesis of two fundamental conceptions of integration: one corresponding to the idea of definite integral as limit of certain approximating sums, and the other to that of indefinite integral understood as a primitive function. It is usual to associate these two conceptions of integration with the names of Leibniz and Newton. In accordance with this distinction (which is largely a matter of convention) we shall call a function of a real variable F indefinite integral, or primitive, of Newton for a function /, if F has everywhere its derivative finite and equal to /. The function / will then be termed integrable in the sense of Newton, and the increment of the function F on an interval I , will be called definite integral of Newton of / on I . As is seen im- mediately, this definition implies that any function which is integrable in the sense of Newton is everywhere finite. This restriction is essential (cf . the example of 7, p. 206) for the unicity of integration in the sense of Newton, which then follows from classical theorems of Analysis, or, if we like, from Theorem 3.1, or from Theorem 7.1 of this chapter. The theory of the integral was first developed on Newtonian lines. This is easily accounted for if we think how much simpler the inverse [ 1] Introduction. 187 of the operation of derivation must have seemed than the notion of de- finite integral as defined by Leibniz. It was A. Cauchy [I, t. 4, p. 122] who returned to the idea of Leibniz in order to apply it to integration of continuous functions, for which the methods of Cauchy and Newton are actually completely equivalent. This equivalence dis- appears, however, as soon as we pass on, with Eiemann, to inte- gration of discontinuous functions. In fact, even in the domain of bounded functions to which the Eiemann process applies, there exist on the one hand (as we see at once) functions which are inte- grable in the sense of Eiemann but have no primitive, and on the other hand (as shown by V. Volterra [1]; cf. also H. Lebesgue [II, p. 100]) functions which have a primitive but are not integrable in the Eiemann sense. Also the Lebesgue process of integration does not include the integral of Newton, not even when the func- tions to be integrated are everywhere finite. Thus, the function F(x) = x* sin (/r/x a ) for x^O, completed by writing )= 0, has in the whole interval [0, 1] a finite derivative which vanishes for x~Q and which is bounded on every interval [e, 1], where 0<e<l. On every interval 0, 1] the function F(x) is therefore absolutely continuous. On the other hand, on the whole interval [0,1] the function is not even of bounded variation. Hence F'(x] is not summable on [0,1], since its indefinite Lebesgue integral could then differ only by an additive constant from F(x) on [0,1], and this is im- possible. We have thus been led to the problem of determining a process of integration which includes both that of Lebesgue and that of Newton. As an application of the method of major and minor func- tions, we shall consider in this chapter ( 6 and 7) the solution of this problem constituted by the Perron integral. Another solution, the Denjoy integrals, will be treated in Chapter VIII. The notions of major and minor functions, and their applications to Le- besgue integration, will be discussed here for arbitrary spaces R m - In defining the Perron integral, however, we shall limit ourselves to functions of one real variable. Although recently various authors have treated the extension of this integral to Euclidean spaces of any number of dimensions, the present state of the theory does not allow us to decide as to the importance of this genera- lization. On the contrary, in the domain of functions of a real variable, the method of major and minor functions as a means of generalizing the notion of integral hag already repeatedly shown its fruitfulness. In the memoir of J. Marcinkiewicz and A. Zygmund [1], the reader will find new applications of this method in connection with certain fundamental problems of the theory of trigonometrical series (cf. also J. Ridder [!!]} 188 CHAPTER VI. Major and minor functions. 2. Derivation with respect to normal sequences of nets. Given a regular sequence 97 ={97*} of nets of intervals (vide Chap. Ill, 2) in a space R m and a function of an interval F in JR mj we shall call upper derivate of F at a point x with respect to the sequence of nets 97 the upper limit of the ratio F(Q)/\Q\ as d(Q)->Q, where Q denotes any interval containing x and belonging to one of the nets of the sequence 97. By symmetry we define similarly the lower derivate of F at x with, respect to the sequence of nets 97. We shall denote these two derivates by (9J) F(x) and (97) F(x). When they are equal at a point a?, their common value will be denoted by (97) F (x) and called derivative of F at x with respect to the sty urnce of nets 31. These definitions are similar to those given in 15, Chap. IV, in connection with derivation of additive functions of a set (93) in a metrical space. It should be observed, however, that additive functions of a set ($) correspond to ad- ditive functions of an interval of bounded variation, whereas in the present we treat derivation of additive functions of an interval without supposing them a priori of bounded variation. For this reason it will be necessary to impose certain restrictions on the nets considered in this , and to distinguish a class of nets which we shall call, for brevity, normal nets. The latter are, in point of fact, the nets occurring most frequently in applications (cf., for instance, ('hap. Ill, p. 58). A system of intervals will be called a normal net in the space U m , when it consists of the closed intervals [a* , a^i ; af\ af+i ; ... ; off*, a**,] for k~ 0, 1, 2,..., which are determined by systems of num- bers aj^ subject to the conditions a^ <j^_, for i = l 9 2, ..., w and k ..., 1, 0, + 1, ..., arid lim a ( > = oo. A regular sequence of ll-+ '00 normal nets will be termed normal sequence. (2.1) Theorem. Let 31 = {97*} be a normal sequence of nets, g(x) a function which is summable in the space K m and F a continuous additive function of an interval such that (i) (97).F(#)> oo at every point Xj except at moat those of an enumerable set, and (ii) ^'(x)^g(x) at almost all the points x at which the function F is derivable in the ordinary sense. Then for every interval /, we have (2.2) i. e. F is a function of bounded variation, whose function of singul- arities is monotone non-negative. [2] Derivation with respect to normal sequences of nets. 189 Proof. Consider the points in every neighbourhood of which there exist intervals / for which the inequality (2.2) is false, and let P denote the set of these points. The set P is evidently closed, and we see easily that the relation (2.2) must hold for every interval I such that 1 C CP. For if this were not the case, we could deter- mine first an interval /CCP such that F(l)<jg(x)dx, and then, / by the method of successive subdivisions, a descending sequence {!} of subintervals of / such that 6(I n )->0 as n-+oo and that F(I n )<il g(x)dx for n=l, 2, .... Therefore, denoting by a the com- i n mon point of the intervals /, we should have a e P, which is clearly impossible. It follows that in order to establish the validity of the in- equality (2.2) for all intervals /, we need only prove that P=0. Suppose therefore, if possible, that P = 0. Let us denote, for any pair of positive integers k and A, by N kth the sum of all the intervals I 00 of the net 9t* for which F(I)>h-\I\. Therefore by writing A T A=//A\/,, A-/! we obtain a sequence {N^} of closed sets whose sum, according to condition (i), covers the whole space except for an at most enumerable get. Consequently, on account of Bairc's Theorem (Chap. II, Theo- rem 9.2), the set P contains a portion which either consists of a single point, or else is contained in a set N h . The former case is excluded since it is evident from the continuity and additivity of the function F that the set P contains no isolated points. Therefore there exists a positive integer h Q and an open sphere S such that O^P-SC-^v Let us write H(I)=F(I) + h .\I\ + f\g(x)\dx where I i is any interval. We shall have H(I)^0 for any interval 1 such that ZCCP, as well as for any interval / belonging to a net 91* of index fc^A and having points of the set N^ in its interior. Therefore ff(I)^0 for any interval I 8 belonging to a net 9U of index fc>A , and consequently, by additivity and continuity of H, we have #(7)^0, i.e. F(I)^ hf\I\ f\g(x)\djr, for any inter- val /CS whatsoever. It follows at once that the function F is of bounded variation in 8 and that the function of singularities of F (cf. Chap. IV, p. 120) is monotone non-negative in S. 190 CHAPTER VI. Major and minor functions. Hence, by condition (ii), F(I)^ f F'(x)dx^ fg(x)dx for every inter- val /C$- But since P-S^O we thus arrive at a contradiction and this completes the proof. As an immediate corollary of Theorem 2.1, we have '(2.3) Theorem. If 97 is a normal sequence of nets in the space K m and if F is a continuous additive function of an interval such that: (i) ^o<(yi)F(x)^( < R)F(x)< + oo for each point x except at most the points of an enumerable set, (ii) the (ordinary) derivative F'(x) is summable on each portion of the set of the points at which this deriva- tive exists ; then the function F is almost everywhere derivable and is the indefinite integral of its derivative. For Theorems 2.1 and 2.3 cf. J. Ridder [2]. Let us remark that in the case where the function F ia of bounded variation, these theorems are included in Theorem 15.7, ('hap. IV, which concerns derivation of additive functions of a set in an abstract metrical space. It follows easily from Theorem 15.12, Chap. IV, that f"(x)=(3l)F'(x) almost everywhere for any regular sequence of nets of intervale 91 and for any additive function of an interval F which is continuous and of bounded variation. This remark enables us to replace condition (ii) of Theorem 2.1 by the following: (ii-fcis) F'(x)^(yi)F'(x)^g(x) at almost all the points x at which the two derivatives F'(x] and (^{)F'(x) exist, are finite and equal. Similarly we may modify condi- tion (ii) of Theorem 2.3. As it follows from an example due to A. J. Ward [7], the jnequality (W)F(x) oo in condition (i) of Theorem 2.1 cannot bereplacedby (^)F(x)> oo. 3. Major and minor functions. Before introducing the fundamental definitions of the theory of the Perron integral, we shall prove (3.1) Theorem. If an additive function of an interval F (not neces- sarily continuous) has a non-negative lower derivate at each point x of an interval /, then F( Proof. Let* he any positive number and write 0(I)=F(I) + e-\I\ for every interval 7. Then (?(,r)>>0 at each point o?/ . Suppose that G(/ )^0. By the method of successive subdivisions, we could then determine a descending sequence {!} of intervals similar to I , such that <?(/)<() for n=0,l,2, ... and that <5(/,,)->0 as n->oo. Therefore, denoting by x the common point of the intervals ?, we shoulgl have <3(.r )^0 which is impossible. Hence (7(/ )>0, and this gives F(I Q )> -|/ ! for each e>0, and finally F(/ )>0. [ 4] Derivation with respect to binary sequences of nets. 191 An additive function of an interval F is termed major [minor} junction of a function of a point / on a figure JS if, at every point x of this figure, o^F(x)^f(x) [+oo:j\(a)</(#)]. It follows at once from Theorem 3.1 that if the functions of an interval U and V are respectively a major and a minor function of a function / on a figure R^ their difference U F is monotone non-negative on jR . (3.2) Theorem. If f is a summable function, then, for each * >0, the function f has an absolutely continuous major function, U and an absolutely continuous minor function V such that, for each interval I, (3.3) 0*^U(I) f(x)dx^e and i Proof. On account of the theorem of Vitali-Carath^odory (Chap. Ill, Theorem 7.6) we c&n associate with the function / two summable functions, one a lower semi - continuous function g and the other an upper semi-continuous function h, such that (i) oo^ g(x)^f(x)^h(x) 4= + 00 at every point x and that (ii) j[g(x) f(x)]djc<c and f[f(x) h(x)]dx<e for every interval/. / / Therefore, if we denote by U and V the indefinite integrals of the functions g and h respectively, we find by Theorem 2.2, Chap. IV, that U*(x)^g(x)^ h(x)^V*(x), and so, on account of (i), that oQ^U^x)^f(x) and +00=^ V s (x)^f(x) at each point x. Finally, (ii) then implies the relations (3.3) and this completes the proof. Theorem 3.2 can easily be inverted. Thus: in order thai a function of a point f be summable, , it is necessary and sufficient that for each e>() there exist two absolutely continuous functions of an interval U and V , the one a major and the other a minor function of /, which fulfil the condition U(I) V (I) < e for erery interval I. (These absolutely continuous functions may clearly be replaced by functions of bounded variation, and if the function / is supposed measurable, then, of course, for its summability there suffices the existence of two functions of bounded variation, one of which is a major and the other a minor function of /.) * 4. Derivation with respect to binary sequences of nets. The theorems of 2 concerned derivation of additive functions with respect to any normal sequence of nets of intervals. For certain purposes however, more special sequences of nets are required. We shall say that a normal sequence {9UU-t,2,. f ncts in ^^ e 8 P aee JR m is binary, if the net 9l*+i (where 4=1,2,...) is obtained by subdividing each interval N of the net Vl k into 2"' equal intervals similar to N. 192 CHAPTER VI. Major and minor functions. An application of this notion may be found in the following theorem which is proved similarly to Lemma 11.8 of Chap. IV: // 9t is a binary sequence of nets, any additive function of an interval F is derivable with respect to 91 at almost all the points at which either (yi)F(x)>oo or (yi)F(x)<+ oo. Another application, of particular interest, is due to A. S. Besicovitch [3] who, by using derivation with respect to a binary sequence of nets, established a theorem on complex functions (vide below 6). The substance of Besicovitch's result is contained in Theorem 4.4 below. We must first, however, give some subsidiary definitions. For definiteness, just as in 11, Chap. IV, we shall fix in the space K m a binary sequence of nets Q={Q*}, where Q* denotes, for fc 1, 2, ..., the net formed by the cubes [ Pl /2*, ( Pl + l)/2*; P J2", (p 2 + l)/2*; ...; pJ2*, (p m + l)/2 k ] where p,, p 2 , ..., p m are arbitrary integers; it goes without saying that in Theorem 4.4 this sequence may be replaced by any binary sequence whatsoever. Given a non-negative number a, we shall say that a function of an interval F fulfils the condition (it) [condition (!")] at a point #, if liminf JP(I)/[(5(I)] a >0 [lim sup F (1)1 [6 (I)]" <0], where / is any interval containing x. If a function /fulfils the condition at every point of a figure /2, we shall say simply that / fulfils this condition on R. Finally, we shall say that a function fulfils the condition (1) at a point, or on a figure, if it fulfils simultaneously the conditions (1^) and (17). We recall further the notation A(E) for the a-dimensional measure of a set E (cf. Chap. II, p. 53). (4.1) Lemma. Given a set E in the space R mj together with a positive integer Jc and a non-negative number a< m, there exists for each >0 a sequence {Q n } of intervals belonging to the nets Q* for fc^fco, which fulfils the folloiving conditions: (i) Z\.t(Qn)Y^(4m) m -[A(E) + *]i n (ii) to each point x of E there corresponds a positive integer k^Jc Q such that all the intervals of the net Q* which contain the point x belong to the sequence {Q n }- Proof. Let us cover E by a sequence {B/}/=,i,2.... of sets such that 0<(}(.E/)<1/2 VH for f=l, 2, ... and such that (4.2) [4] Derivation with respect to binary sequences of nets. 193 Let us denote by fc,, for each t=l,2, ..., a positive integer such that (4.3) l/2 i We easily see that fc/>fco for every i, and that each net Q* y , for i=l,2, ..., can contain at most 2 m intervals having points in common with EI. Let {#}_, 2 be the sequence of all the intervals belonging to the nets QA^ Q* 2 , ..., Q* /? ... and having points in com- mon with the sets E v E^....E l9 ... respectively. The sequence {Q n } clearly fulfils the condition (ii). Moreover, we find on account of (4.3), ^^ 71 I / I and this by (4.2) gives at once the condition (i). (4.4) Theorem. Suppose that F is a continuous additive function of an interval in the space R m and fulfils the condition (1) where O^a < m, and that g is a summable function. Suppose further that (i) (Q)F(x) > oo at every point x except at most those of a set E expressible as the sum of an enumerable infinity of sets of finite measure (A), and that (ii) (Q)F(x)^ g(x) at almost all points x; then (4.5) F(I Q )^[g(x)dx / for every interval 7 . Proof. Since the function F is continuous, it will suffice to prove (4.5) in the case in which the interval I belongs to one of the nets Q*, to the net Q* , say. Further by changing, if necessary, the values of g on a set of measure zero, we can assume that the inequality (Q)F(x) ^g(x) holds at every point x. Let e be a positive number and let F be a minor function of g (cf. 3, particularly Theorem 3.2) such that (4.6) Let us write G(I)=F(I) V(I)+e-\I\, where / denotes any interval. We shall have ())<?(#) ^(Q)F(x) V(x) + f ^f >0 jit every point x except at most at the points of E. Finally, since Fs(#)< + at every point 07, the function V fulfils the condition (17) and the func- tion therefore fulfils the condition (1^). 194 CHAPTER VI. Major and minor functions. Let us now represent the set E as the sum of a sequence {Ei}t=\, 2,... of sets of finite measure (A a ), and denote, for each pair of positive integers i and n, by R iifl the set of the points x such that the inequality 0(I)> e-[d(I)] a /2 i [A a (E i ) + I] holds whenever I is an interval containing x and belonging to one of the nets Q* for Tc^n. The sets R iin are evidently measurable (93) (they are actually sets ((&<$)) Moreover, since the function G fulfils the condition (1^), .the sum Rt,n must, for each integer i, cover the whole space JK m . Hence, n writing E itn =E r (Rt, n Rt, n -i) for n>l, and E it i = E r R ttl , we find that (4.7) A n (E t ) = ^A a (E lt ^ for <=1, 2, ... . 71 This being so, it follows from Lemma 4.1 that for each pair of positive integers i and n, we can determine a sequence {Qfy J==l 2 of cubes which belong to the nets Q* for k^n, and fulfil the fol- lowing conditions: (4.9) to each point XE itn there corresponds an integer lc^k Q such that each cube of the net Q A , containing X, belongs to the se- quence {Q ( f n ) Jsslt2tt j (4.10) each cube Q^ n has points in common with the set E f n and therefore fulfils the inequality G(Q ( ^)>-e.[6(Q ( ^)]/2 i .[A a (E l ) + ll For brevity, let us agree to say that an interval has the prop- erty (A), when it is representable as the sum of a finite number of non-overlapping intervals I each of which either fulfils the in- equality (?(I)>0, or else coincides with one of the cubes Q ( ^ n . We remark that on account of (4.10), (4.8) and (4.7), the inequality t,n,j t,n is valid whenever R is a figure consisting of any finite number of non-overlapping cubes Q ( f J) nJ and therefore that the inequality <?(!)>- (4m) m -e must hold for every interval I having the prop- erty (A), [f 5] Applications to functions of a complex variable. 195 We shall now show that the interval I itself has the prop- erty (J.), so that O(I )^ ~(4:m) m e. Let us suppose the contrary. We could then, starting with / , construct a decreasing sequence {I?} of cubes belonging to the nets Q* and none of which has the property (A). Let x be the common point of these cubes. Then either x^eE, and consequently, by (4.9), the sequence contains cubes Qfy or x eCE, so that (Q)G(x ) >0, and therefore G(I P )>Q for each sufficiently large p. Thus in both cases, the sequence {I p } would contain intervals with the property (A) and we arrive at a contradiction. It follows that G(I )> (4m) /n f, and therefore, by (4.6), that since e is an arbitrary positive number, this gives the relation (4.5). * 5. Applications to functions of a complex variable. We now interpret the points of the plane M 2 as complex numbers and, as usual, we call complex function of a complex variable every function of the form u-\-iv where u and v are real functions in the whole plane, or in an open set. The functions u arid v are termed real part and imaginary part of the function /. A complex function is said to be continuous (at a point, or in an open set), if its real and imaginary parts are both continuous. Given a complex function /, continuous m an open set 6r, and having the real and imaginary parts u and v respectively, we shall write for every interval / = [!, b^ a 2 , b 2 ] contained in G: Mi ( 5 -D J 2 (/; 1)= I l u ( b v .y) M(I, y)] rf y /[(* * 2 ) ( 2 )] rf-p. fl.; ^/l and The expression J(/; I), which will also be denoted by j f dz, will (/) be called curvilinear integral of the function / along the boundary of the interval /. The function / will be termed holomorphic in an open set (?, if J(/;/) = for every interval 1(^0. (The equivalence 196 CHAPTER VI. Major and minor functions. of this definition of the term ,,holomorphic" used here in place of terms such as "regular", "analytic", etc. with the more familiar definitions of the theory of complex functions, follows from the well-known theorem of Morera [1].) We verify at once that this relation holds when f(z) az + b where a and b are any complex constants. If / is a complex function, continuous in an open set (?, the expressions J\(/; I) and J 2 (/; /) are continuous additive functions of the interval 1 in (?. Moreover for each interval I in O. On account of Theorem 2.3, we there- fore obtain at once the following theorem due to J. Wolff [1] (cf. also H. Looman [2] and J. Bidder [1; 2]): (5.2) Theorem. A complex function /, continuous in an open set G, is holomorphic in O if at almost all points z of (?, iminf^U C <XQ)-M> |V| U liminf f(z)dz and if at all points z of (?, except at most those of an enumerable set, limsup-j^yr I f(z)dz where Q denotes cmy square containing z. A complex function is called derivable at a point z , if the ratio [/(*) f( z o)]/( z z o) tends to a finite limit when z tends to z in any manner. This limit is called derivative of / at z and is denoted by f(z Q ). Let / be any complex function, defined in the neighbourhood of a point z . If we have lim sup \[f(z + h)~ f(z Q )]/h\< + oo, we can write f(z)=f(z Q )+M(z)'(z z ), where M(z) is a function of z which is bounded in the neighbourhood of z ; and we then easily find that the ratio |J(/;Q)|/|Q|, and a fortiori the ratios |Ji(/;<?)|/|<?| and |J 2 (/5 Q)\/\Q\i must remain bounded when Q denotes any suf- ficiently small square containing z . If, further, the function / is derivable at z , we have f(z) = f(z Q )+f '(ZQ)-(ZZQ) + I(Z)'(ZZQ), where |e(z)|->0 as z->z^ and the ratios in question tend to zero as <$(#)-> 0. Finally, let us observe that if the function / is continuous, the expressions J x (/; I) and J a (/; I), considered as functions of the [5] Applications to functions of a complex variable. 197 interval Z, both fulfil the condition (1J of 4. Therefore, if we apply Theorem 4.1, we obtain the following theorem due to A. S. Besi- covitch [3] (cf. also S. Saks and A. Zygmund [2]): (5.3) Theorem. A complex function /, continuous in an open set (r, is holomorphic in G if it is derivable at almost all the points of G and if further lim sup \[f(z + h)~ -f(z)]/h\ < + oo at each point z of G except at /i-K) most those of a set which is the sum of a sequence of sets of finite length. The theorem of Besicovitch may be regarded as a generalization of the classical theorem of E. Goursat (!]: A complex junction /, continuous in an open set G, is holomorphic in G if it is everywhere derivable in G.T. Pompeiu [1] showed that it is enough to suppose / derivable almost everywhere, provided that we restrict the expression lim sup \[f(z-\-h] f(z)]/h to be bounded in G. Finally, h-+Q H. Loornan [3] (cf. also J. Ridder [2]) replaced the condition that the expres- sion lim sup (f(z +h) f(z)]/h\ is bounded by the condition that this oxpres- /?-M) sion is finite at each point of G. Theorem 5.3 evidently includes all these generalizations. The theorems of Morera and of Goursat, and their generali- zations furnished by Theorems 5.2 and 5.3, contain criteria for holomorphism which are based on the notion of curvilinear integral and of derivation in the complex domain. The classical theorem of Cauchy is an instance of a criterion of a different kind, expressible in terms of real variable conditions on the real and imaginary parts of a complex function; we have in fact, according to this theorem: in order that a continuous function of a complex vari- able f(z) = u(Xj y) + iv(x, y) be holomorphie 'in an open set (?, it is necessary and sufficient that the partial derivatives u x , u^ v' x , v y should all exist in G and be continuous, and that they everywhere fulfil the Cauchy- Biemann equations u' x =v , and u=v x . A series of researches begun by P. Montel [1] has been de- voted to the reduction of these conditions, particularly that of the continuity of the partial derivatives. The problem was finally solved by H. Looman [2] and D. Menchoff (vide the first ed. of this book, p. 243, and D. Menchoff [I]) who succeeded in removing completely the condition in question without replacing it by any other. It is remarkable that a classical problem of such an elementary aspect should only have been solved by a quite essential use of methods of the theory of real functions. 198 CHAPTER VI. Major and minor functions. (5.4) Lemma. Let w be a real junction of one variable, derivable almost everywhere in an interval [a, 6]; let F be a closed non-empty subset of this interval, and let N be a finite constant such that whenever x t eF and x 2 e [a, b]. Then (5.5) w(b)w(a)fw'(x)dx ^N-(ba\F\). F Proof. Let us denote by F l the set obtained by adding the points a and b to the set F. The function t0, equal to w on F l and linear on the intervals contiguous to F 19 is evidently absolutely continuous on [a, b] (and even fulfils the Lipschitz condition). Hence b (5.6) w(b)w(a) = w(b) w(a) = fw(x)dx. Now w'(x)--=w'(x) at almost all the points x of F and |0' at each point x outside F. The relation (5.5) therefore follows at once from (5.6). (5.7) Lemma. Let w(x,y) be a real function whose partial derivatives with respect to the two variables x and y exist at every point of a square Q, except at most at the points of an enumerable setj and let F be a closed non-empty subset of Q, and N a finite constant such that 2 Xj\ and \w(x l9 y 2 ) whenever (s^y^eF, (#2>3/i)e#> wwi (x^y^eQ. Then if [a^ b^ a 2 , 6 2 ] denotes the smallest interval (which may be degenerate) containing F, we have (5.8) -f fw x (x, y)dxdy F Proof. Let us choose arbitrarily two points (a?', a 2 ) and (a?", & 2 ) belonging to the set F and situated on the two sides of the interval [0^ brf a 2 , b 2 ] parallel to the rr-axis. For any point of [a^ b^ we havelw^ft,)-!^^ + \w(x',b 2 )w(x',a 2 )\ + \w(x',a 2 ) w?(^,a 2 )|; and hence, denoting by I [5] Applications to functions of a complex variable. 199 the length of the side of the square #, we obtain < J y.[| a? "~|+ x '- x "\ + \a 2 -b 2 \ + \- We now denote for any point of [%, 6J, by F$ the set of all the points y of [a 2 , fc 2 ] such that (, y)e-F. Let J. be the set of the points of the interval [a 1? fcj for each of which F^Q, and let B denote the set of the remaining points of [%, fr t ]. On account of Lemma 5.4 we have whenever eA, and if we integrate the two sides of this inequality with respect to on the set J., we find On the other hand if we integrate (5.9) with respect to on the set H, we obtain | f\w(, & a ) w(f,a 2 )]d|< 4NI- \B\ <4#. \QF\, B and by adding this to (5.10) we obtain the first of the inequalities (5.8). The second inequality follows by symmetry. (5.11) Theorem of Looman-Menchoff. If the functions u(x,y) and v(x,y), continuous in an open set (7, are derivable with respect to x and with respect to y at each point of G except at most at the points of an enumerable sretj and if u' x (x,y)~v'y(x,y) and u y (x,y) = v x (Xjy) at almost all the points (X, y) of 6, then the complex function f=u + iv is holomorphic in G. Proof. Let us denote by F the set of the points (x, y) of G such that the function / is not holomorphic in any neighbourhood of (a?, y). The set F is evidently closed in G and the function / is holomorphic in GF. It thus has to be proved that F is empty. Suppose therefore, if possible, that F ^ and let F n denote, for each positive integer n, the set of the points (a?, y) of G such that, whenever |ft|^l/n, none of the four differences u(x + h,y)u(x,y) u(x,y+h)u(x,y), v(x+h,y)~-v(x,y), v(x,y + h)v(x,y) exceeds \nh\ in absolute value. By continuity of the functions u and v, each of the sets F n is closed in G. On the other hand, the sets F n cover the whole set G, except at most an enumerable set consisting of 200 CHAPTER VI. Major and minor functions. the points at which the functions u and v are not both derivable with respect to x and with respect to y t simultaneously. Therefore, on account of Baire's Theorem (Chap. II, Theorem 9.1), the set F(2 O contains a portion which either reduces to a single point, or else is contained entirely in one of the F n . The former possibility is ruled out, since, as we easily see on account of the continuity of /, the set F cannot contain any isolated points. There must therefore exist a positive integer N and an open sphere S such that Q^= Let Q be any square contained in 8. such that and Q-F^O. We denote by I = [a v b^ a 2 , & 2 1 the smallest interval con- taining Q-F. By applying the evaluations of Lemma 5.7 to the integrals on the left-hand sides of the formulae (5.1) and by taking into account the relations u x (x,y)v' lj (x,y) and u y (x,y) = v' x (x,y) which are, by hypothesis, fulfilled almost everywhere, we find iJ^/jjOKlOJV-lC F and |J 2 (/; I)\^1QN.\Q~ F , and therefore |J(/; /)j^20^-|# JF|. This last inequality may also be written |J(/5 01^20^-10 J 1 ), since the figure QQI contains no points of F in its interior, and since therefore J(/;Ji) for each interval R contained in $01. Now let 2 (# ?#o) be any point of $, and let Q be any square containing z . By what has just been shown, if z Q eF we have |J(/;0)I/I0K20JV-|0 F\I\Q\ as soon as <$(#)< W; the ratio J(f>Q)l\Q\ therefore remains bounded as 6(Q)->0 and tends to zero whenever z is a point of density of F. Further |J(/; Q)\I\Q\~>Q as d($)->0, whenever z eSF, since J(f}Q) = Q for every square Q which does not contain points of F. Therefore by Theorem 2.3, the function / must be holomorphic in S. This is, however, excluded since S-F^Q. We thus arrive at a contradiction and this completes the proof. Theorem 5.11 was stated (even in a more general form) by P. Mont el [2] as early as 1913, but without proof. The proof supplied by H. Looman [2] in 1923 was found to contain a serious gap which was only finally filled in by D. Men- choff (cf. D. Menchoff [I] and the first edition of this book, p. 243). By making use of general theorems on derivates (vide, below, Chap. IX) it is possible to weaken slightly the hypotheses of the theorem. Thus instead of assuming partial derivabih'ty of the function u and v, it is sufficient to sup- pose that at each point of (except at most those of an enumerable set) these functions have with respect to each variable, x and y, their partial Dini derivates finite. This condition implies (cf. Chap. VII, 10. p. 236, or Chap. IX, 4) partial derivability of the functions u and v with [6] The Perron integral. 201 respect to each variable at almost all points of G (this generalization of the theorem of Looman-Menchoff does not require any alteration of the proof; for other and much deeper generalizations, vide the memoirs of D. Menchoff [1: 2]). The extension of Theorem 5.11 which we have indicated, includes in par- ticular the theorem of Looiuan mentioned above, p. 197, but not however the theorem of Besicovitch (5.,'J). It would be interesting to establish a theorem which would include both the theorem of Besicovitch and that of Looman-Menchoff. 6. The Perron integral. For functions of one real vari- able, as announced in 1, the method of major and minor func- tions leads to an important generalization of the Lebesgue integral. A function of a real variable, /, is termed integrable in the sense of Perron, or ^-integrable, on a figure R in /fj, if 1 / has both major and minor functions on A' , and if 12 the lower bound of the numbers V(K Q ), where V is any major function of / on K QJ anil the upper bound of the numbers F(/? ), where V is any minor function of /, are equal. The common value of the. two bounds is then called definite Perron integral, or definite ^-integral, of / on K OJ and de- noted by (&) I f(x)dx. It is evident that for F-integrability of a June- #0 lion f on a figure ff it is necessary and sufficient that for each t >0 there should exist a major function U and a minor function V of f on R Q such that U(R )V(K Q )<f. Since (of. 3, p. 190) the function U V is monotone non- decreasing for every major function U and every minor function V of /, it follows that every function which is ff-integrable on a figure R Q , is so also on every figure R R (} . The function of an interval P(I)~(&) j f(x)dx, thus defined for every interval /C#o> i g called / indefinite Perron integral, or indefinite ff-integral, of / on R . As we see at once, P(I) is an additive function of the interval /. Moreover, given any positive number f, there exist always a major function U and a minor function V of /, such that 0^t/(7) P(/)^f and O^P(/) V(I)^e for every interval 1C ^o> ail( l since U(x)> o and V(x) < + at each point ,r of 72 , it follows at once that the function P is continuous. Just as in the case of the Lebesgue integral, a function of a real variable is termed indefinite ^-integral [major function, minor function] of a function /, if this is the case for the function of an interval determined by it (cf. Chap. Ill, 13). 202 CHAPTER VI. Major and minor functions. As we see at once from Theorem 3.2, every function which is integrable in the sense of Lebesgue on a figure JB , is so in the sense of Perron, and its definite Lebesgue and Perron integrals over R Q are equal. On the other hand, if F is the primitive of Newton (cf. 1) of a function /, the function F is at the same time a major and a minor function of /, and therefore is the indefinite ^-integral of /. It follows that Perron's process of integration includes both that of Lebesgue and that of Newton. We shall establish some fundamental properties of the Perron integral. (6.1) Theorem. Every &-integrable function is measurable, and is almost everywhere finite and equal to the derivative of its indefinite integral. Proof. Let / be a function of a real variable, <^-integrable on an interval / , and let P be its indefinite ^-integral on I . It has to be proved that the function P has at almost all points a?, a finite derivative equal to f(x). For this purpose, let be any positive number and U a major function of / such that (6.2) 7(/ )-P(I )<A Let us write H=UP. The function H, as monotone non-decreasing, is almost everywhere derivable, and if we denote by E the set of the points x of Z at which H'(x)^e, we find, by (6.2) and Theo- rem 7.4, Chap. IV, that \E\<f. Now at each point xeI where the function H is derivable, U(x)=H'(x) + P(x); hence P(x)> oo and P(x)^U(x)-~ e>/(#) e at almost all the points x of I E. Therefore, since |jE|<l, e being an arbitrary positive number, it follows that o^P(x)^ f(x) at almost all the points x of I . By symmetry this gives also -f oo^P(x)^f(x), and finally oo^P'(x)=f(x) almost everywhere in I . (6.3) Theorem. If two functions f and g are almost everywhere equal on a figure R and one of them is ff'-integrable on R Q , so is the other and the definite ^-integrals of f and g over R are equal. Proof. Suppose that the function / is ^-integrable and denote by A the value of its definite integral over R Q . Let be any positive number and let U and V be two functions of an interval, which are respectively a major and a minor function of / on R and which fulfil the inequalities [ 7] Derivates of functions of a real variable. 203 (6.4) U(R )^A^V(R Q ) and Let us denote by E the set of the points x at which The function equal to +00 at all the points of E and to every- where else is therefore almost everywhere zero, and by Theorem 3.2 has a major function such that 0^{?(# )^e/3. We have G(jc)= -foo at each point xeE and writing U l =U+0 J V^ = V ff, we see that the functions of an interval U l and V l thus defined are respectively a major and a minor function of the function g on R Q . Moreover by (6.4), U^R^^A^V^R^ and U^B^ V^JR^^e. Therefore the function g is c/^integrable on J? and A^(&) f g(x)dx, which completes the proof. * (6.5) Theorem. Every function f which is tf~integrable and almost everywhere non-negative on a figure /? , is summable on this figure. Proof. We may assume, by Theorem 6.3, that the function / is everywhere non-negative on 7? . Therefore if U is any major function of /, we have U(x)^f(x)^0 at every point #el? , and consequently, by Theorem 3.1, the function U is monotone non- decreasing. Its derivative U '(x) is therefore summable on /? , and, since U'(x)^f(x)^0 almost everywhere, the function / is also sum- mable on R . Theorem 6.5 shows thai, although Perron integration is more general than Lebesgue integration, the two processes are completely equivalent in the case of integration of functions of constant sign. 7. Derivates of functions of a real variable. Certain of the theorems of 2 and 3 can be given a more complete statement when we deal with functions of one real variable. We shall begin with the following proposition which is due to Zygmund: (7.1) Theorem. If F(x) is a finite function of a real variable such that (i) lim sup^(# ftXJP(a?)^limsupjP(j? + h) at every point x, and (ii) the set of the values assumed by F(x) at the points x where contains no non-degenerate interval, then the function F is monotone non-decreasing. Proof. Suppose, if possible, that there exist two points a and b such that a<b and that F(b)<F(a). Then, denoting by E the set of the points x at which F^(x)^.Q, we can determine a value y not belonging to the set F[E] and such that F(b)<y <F(a). Let 204 CHAPTER VI. Major and minor functions. Xy be the upper bound of the points x of [a, 6], for which We shall obviously have a<x Q <b, F(XQ)=I/ Q , all( l ^(^)^#o f r each point x of the interval [x^b]. Therefore F^(x^)^0, although X Q does not belong to E. This is in contradiction with the definition of the set E. Lot us mention the following consequence of Theorem 7.1: Dini's Theorem. Given on an interval I=[a, 6] a continuous function F(x), the upper and lower bounds of each of its four Dini derivates arc respectively equal to the upper and lower hounds of the ratio - , where x } and x 2 are any t r *^9'~ '^1 points of f. Let, for instance, m be the lower bound of the derivate F~*~(JT) on the inter- val /, and suppose first that m> oo. Then, if m' denotes any finite number less than m, the function -F(x) m'*x has everywhere on [a, />] its upper right- hand derivate positive; and so by Theorem 7.1. F(x 2 ) ^'(^i) *tn''(jc 2 ^iK and therefore also [F(x 2 ) F(jr l )]/(x 2 xj"- w t for every pair of points x l and x. t of / such that x } <s 2 . Since the inequality just obtained is trivial in the case- rn --oo, the theorem follows. An immediate consequence is the following theorem: // any one of Ihe four Dini derivates of a continuous function is continuous at a point, so are the three others, and all four der irate* in question are equal, so that the function considered is derivable at this point. These two propositions were proved by IT. Dini [I] in 1878. (7.2) Theorem. If H is a finite function of one variable such that (i) lim sup H(x h) ^ H (x) ^ lim sup H(x + h) at every point x, and A-M) + /i-H)-h (ii) H + (x)^Q at every point x except at most at those of an enumer- able set, then the function H is monotone non-decreasing. Proof. Let f be a positive number and write F(x) = H(x) + ex. We have F^(x)^e>0 at each point x except at most at those of a finite or enumerable set E. The set F[E] being, with jE, at most enumerable, it follows from Theorem 7.1 that the function F(x)^H(x) + fx is non-decreasing for each e>0; and by making ->0 we obtain the assertion of the theorem. (7.3) Theorem. Suppose that F is a continuous function and g a &-integrable function of a real variabUj and that, further, we have (i) F*(x)^g(x) at almost all points x and (ii) F*(x)> oo at every point x, except at most at those of an enumerable set; then (7.4) for every pair of points a and b such that a<b. [7] Deriyates of functions of a real variable. 205 //, in addition, (ij F(x)^g(x)^F^~(x) at almost all points x and (iij) F*(x) > oo and F^(x) < + oo at every point x except at most at those of an enumerable set, then the function F is an indefinite ^-integral of g. Proof. We may obviously assume that F^(x)^g(x) at every point x. Therefore, denoting by V any minor function of </, and writing H=FV, we shall have H + (x)^F~*~(x) V+(x)^0 at every point x, except at most at those of a finite or enumerable set where F+(x)== oo. Further, since the function F is continuous, the inequality V(x)<+ oo, which holds at every point x, implies that the function H satisfies the condition (i) of Theorem 7.2. Consequently. by Theorem 7.2, H(b)H(a)^0 J i.e. F(b)F(a)^V(b)V(a), and since V is any minor function of g, we obtain the inequality (7.4). The second part of the assertion is an immediate consequence of the first part. As we easily see, the condition of continuity of the function F in the first part of Theorem 7.3 may be replaced by the condition (i) of Theorem 7.1. Theorem 7.3 constitutes, on account of Theorem 7.4, Chap. IV, a general- ization of the following theorem of Lebesgue fl, p. 122; 2; 3; 4; II, p. 183]: in order that one of the derivates of a continuous function, supposed finite, be sum- mable, it is necessary and sufficient that this function be of bounded variation; its absolute variation is the integral of the absolute value of the derwate in question. Let us add that in the case in which the function F is assumed to be of bounded variation, Theorem 7.3 is included in Theorem 9.6 of Chap. IV. The condition (ii) of the first part of Theorem 7.3, as well as the con- dition (ii,) of the second, is quite essential for the validity of the theorem. It is possible, in fact, to give an example of a continuous function whose derivative exists everywhere and is summable, without the function being the indefinite integral of its derivative, and this because the latter assumes infinite values. To see this, we shall first show that given any closed set E of measure zero in an interval 7 =[a, b] t there exists a function G, absolutely continuous and in- creasing in J , which has a derivative everywhere in J and fulfils the conditions (7.6) '(x) = -foo for xeE and G'(x)^+oo for xeJ^E* Let us suppose for simplicity that E contains the end -points a and 6 of J and let us denote by ([an.6fi]} the sequence of the intervals contiguous to E. Let {h n } be a sequence of positive numbers such that 00 (7.6) \imhn/(b n a/,)=+oo and (7.7) ^X = l n =-! 00 (it suffices to write, for instance, h n ~\r n \'r n+ ^, where r n=*-r - /* (&/</)) T , ., /=/i Let us write /7\ nt*\-\ M* a*) 1 '* (b n -x) V2 , when a /1 <.r<6 n , l/.OJ y{X)i \ -hoo, when xcE. 206 CHAPTER VI. Major and minor functions. Thus defined the function g(x) is non-negative on <7 and summable on t/ 0> bn b since ^g(x)dxjiUn t so that by (7.7) we have g(x)dx=7i. Let be the indefi- a n a nite integral of g on J . In order to verify that the function Q fulfils the con- ditions (7.5), we observe that the function g(x) is continuous for every xeJ E; on the other hand, if we denote by m n the lower bound of g(x) in \a n , b n ] we derive from (7.6) that lim m rt =lim 2h n /(b n a,,)=-f oo, from which it follows that n n lim g(x)~ + oo=g(x Q ) for every x eE. Consequently Q'(x)g(x) for every x, X-+XQ and therefore, by (7.8), the conditions (7.5) hold. Now let (cf. Chap. ITI, (13.4)), H(x) be a continuous non -decreasing sin- gular function on J , which is constant on each interval contiguous to the set E, and such that H(a)^H(b). Let us put F=G+H. As we verify easily from (7.5), we have F'(x)G'(x) = g(x) at every point x of J . The function F therefore has everywhere a derivative which is summable on J . But, since H is the function of singularities of the function F, the latter is certainly not absolutely continuous, let alone the indefinite integral of its derivative. (The functions O and F provide at the same time an example of two functions whose deriv- atives, finite or infinite, exist and are everywhere equal, without the difference Q F being a constant; cf. H. Hahn [1] and S. Ruziewicz [1].) In connection with these examples, it may be interesting to mention the following theorem (vide G. Goldowsky [1] and L. Tonelli [8]): (7.9) Theorem. If a continuous function F has a (finite or infinite) derivative at each point of /^ except perhaps at the points of an enumerable set, and if this derivative is almost everywhere non-negative, the junction F is monotone non-de- creasing. Proof. Let E be the set of the points x such that the function F is not monotone in any neighbourhood of x. The set E is evidently closed, and the function F is non -decreasing on every interval contained in CE. It therefore has to be proved that the set E is empty. Suppose, if possible, that E^O, and denote for every positive integer n by P n the set of the points a; for which the inequality 0<a? x<l/n implies F(x'}~ JF(x)< (x' x) however we choose x'. Similarly let Q n be the set of the points x for which the same inequality implies F(x') F(x)^2(x' x). We see easily that the sets P n and Q n are closed, and that they cover the whole straight line JBj except at most the finite or enumerable set of the points at which the function F is without a derivative. Consequently, by Baire's Theorem (Chap. II, Theorem 9.2) the set E must contain a portion which either 1 reduces to a single point, or else 2 is contained in one of the sets P n , or finally 3 is contained in one of the sets Q n . The first case is obviously impossible, since the set E has no isolated points. Let us therefore consider case 2, and suppose that there exists a positive integer n and an open interva! / such that Q^E*I_P n(> . We may clearly suppose that 6(1) < l/n . Since by hypothesis, F'(x)^-Q almost everywhere, the set P/I O is certainly non -dense. Let [a, 6] denote any interval contiguous to E*I. The function F is then non-decreasing on [a, 6] and this contradicts the fact that, since o and 6 belong to P/^ and 6 a< l/n , we have F(b) F(a)^(b a)<0. [81 The Perron-Stieltjes integral. 207 There now remains only case 3. In this case there exists an open interval / such that the set E*I is non-empty and is contained in one of the sets Q n . But then F^(x)^~ 2 everywhere in 7, and F'(sc0 almost everywhere, in /. Therefore, by Theorem 7.3, the function F is non -decreasing in 7, and this again is impos- sible since toe interval 7 contains points of E in its interior. We thus arrive at a contradiction in each of the three cases, and this proves our assertion. Let us mention a corollary of Theorem 7.9: If F is a continuous function having a derivative at every point, except per- haps at those of an enumerable set, and if there exists a finite constant M such that \F'(x)\^M at almost all points x, then the function F is the indefinite integral of its derivative. 8. The Perron-Stieltjes integral. Among the various gen- eralizations of the Stieltjes type for the Perron integral (vide for in- stance R. L. Jef f ery [2; 3], J. Bidder [9] and A. J. Ward [3]), that due to Ward has the advantage of including the others and of defining the process of Stieltjes integration with respect to any finite func- tion whatsoever. In this we shall give the fundamental defi- nitions and results of the theory of Ward. For a deeper analysis, in the case in which the function with respect to which we integrate is of generalized bounded variation in the restricted sense (vide below, Chap. VII) the reader should consult the memoir of Ward referred to. As in the two preceding we shall consider only functions defined in JB X , i. e. functions of a linear interval or of a real variable. We shall, moreover, restrict ourselves to integration of finite func- tions. This restriction is essential for the methods which we shall employ. Given two finite functions / and (?, an additive function of an interval U will be termed major function of / with respect to on an interval Z , if to each point x there corresponds a number f>0 such that U(I)^f(x)G(I) for every interval / containing x and of length less than t. The definition of minor junction with respect to O is symmetrical, and by following the method of 6, p. 201, with the help of the notions of major and minor functions with respect to Oj we define Perron- Stieltjes integration, or PS-integration with respect to any finite function O whatever. The ^-integral of a func- tion / with respect to a function on an interval Ifi=[a,b] will t> be denoted by (0S) ff(x)dG(x), or by (fS) ff(x)dG(x). 208 CHAPTER VI. Major and minor functions. If U and V are respectively a major and a minor function of the same function / with respect to the same function (?, their difference V V is evidently monotone non-decreasing. The criterion for ^-integrability of a function is entirely similar to that for {/Mntegrability given in 6, p. 201, and it follows that every func- tion which is <?-integrable on an interval I , is so equally on each subinterval of / . We are thus led to the notion of indefinite PS-integral with respect to any finite f unction (?. This indefinite inte- gral is an additive function of an interval, and is continuous at each point of continuity of the function G. Finally we observe that the ^-integral possesses the distributive property which we may express as follows: If each of the two finite functions / x and / 2 is &$-integrable on an interval I Q with respect to each of the two functions G 1 and (? a , then each linear combination of the functions / x and / 2 is <PS-integrable with respect to each linear combination of the functions (TJ and G 2 , and we have for all numbers a 1? a 2 , b^ and b 2 . If G(x) = x for every point x (or, what amounts practically to the same, if G(I)=\I\ for each interval /) ^-integration with respect to G coincides with ^-integration. In fact, if / is any finite function, each major [minor] function of /with respect to the function G(x)= x in the sense of Ward, is at the same time a major [minor] function of / in the sense of the definition of 3; the con- verse is not true in general, but we see at once that if U is a major function of / in the sense of 3, the function U(x) + ex is for each >0 a major function of / with respect to G(x) = x. Thus the Perron-Stieltjes integral includes the ordinary Perron integral, at any rate as regards integration of finite functions. On the other hand, the Perron-Stieltjes integral includes also the Lebesgue- Stieltjes integral. We have in fact (8.1) Theorem. A finite function f integrable in the Lebesgue-Stieltjes sense on an interval /oC^o^ol w ^ Aspect to a function of bounded variation 6, is so also in the Perron-Stieltjes sense and we have r r' (8.2) (^)/tt?=/*M/(^^ [8] The Perron-Stieltjes integral. 209 Proof. Let us denote for brevity, by A the right-hand side of the relation (8.2). We may evidently assume that the function / is non-negative and it is enough to consider only the following two cases: 1 is a continuous non-decreasing function. The proof is then just as in Theorem 3.2. Let be any positive number. Since the function / is finite, there exists by the theorem of Vitali- Carath6odory (Chap. Ill, 7) a lower semi-continuous function gr, integrable (G) in the Lebesgue-Stieltjes sense, such that g(x)>f(x) al each point x and such that | [?(#) f(x)]dG(x)<e. Denoting by C /o the indefinite integral (0) of the function </, and taking account of the lower semi-continuity of </, we see easily that U is a major function of / with respect to G on Z . Moreover, the function G being continuous by hypothesis, the number A is equal to the integral 9 and we find 0^[7(I ) A<e. By symmetry we determine r fdG also a minor function V of / with respect to G on J in such a manner that O^A F(I )<, and this establishes ^-integrability of/ on I and at the same time the validity of the formula (8.2). 2 G is a non-decreasing saltus-function. Let us denote by \x n }n=-\,2 t ... the sequence of the points of discontinuity of G which are in the interior of the interval I ; and let e be any positive number and {A fl }n=i,2 f ... a sequence of positive numbers such that (8.3) ykn'[G(x n +) G(x n ~)]<e and lim^ = + oo. ^* n n Let us define a function h in R^ by writing: h(x) = f(x) for all the points x of I which are distinct from the points #; h(x n ) =f(x tl )+ k n for n=l, 2, ...; and h(x)=f(a Q ) for x<a oj and h(x)=f(b ) for x>b Q . Finally let us write, for each interval ! = [, 6], b U(I)=fh(x)dO(x)-{h(a)[G(a)~G(a-)] + h(b)[G(b+)-G(b)]}. a The function of an interval U thus defined is evidently ad- ditive, and as we easily verify, is a major function of / with respect to G on I . Moreover, it follows at once from (8.3) that 0^ U(I Q ) A^e. Similarly we determine a minor function V of / with respect to G so as to have O^A V(I )^e; hence A = (&S) f fdG, and this completes the proof. /u 210 CHAPTER VI. Major and minor functions. Formula (8.2) brings out the fact that the definite Perron -Stieltjes and Lebesgue-Stieltjes integrals are not always equal, even for a function / integrable in both senses. This is due to the fact that the indefinite integral of Lebesgue- Stieltjes is not in general an additive function of an interval. We could, of course, modify the definition of this integral so as to ensure its additivity as a function of an interval. The term in brackets { } would then disappear from the formula (8.2), but it would then be necessary to give up the additivity of the indefinite Lebesgue-Stieltjes integral considered as a function of a set (cf. Chap. VIII, 2). Let us mention further the following generalization of Theo- rem 6.1 on derivation of the indefinite Perron integral: (8.4) Theorem. If P is an indefinite PS-integral of a finite function f with respect to a function (?, then, at almost all points x, the ratio (8.5) [P(I)-f(x)G(I)]j\I\ tends to as (5(7) ^0, where I denotes any interval containing x. Hence at almost all points x, P(x)=f(x)G(x) and P(x)=f(x)G(x) or else P(x) = f(x)G(x) and P(x) = f(x)G(x) according as f(x)^Q or f(x)^.Q] in particular P'(x) at almost all points x where f(x) = Q. Proof. The proof is quite similar to that of Theorem 6.1. Let I Q be an interval, e a positive number, and U a major function of / with respect to G on J such that U(I Q ) P(I )<e 2 . We write H=U~P. The function H is monotone non-decreasing, and we have H'(x) < at every point x f I Q except at most those of a set E of measure less than e. Now, since U(I) /(#)(?(!) >0 for every point x and for every sufficiently small interval I containing #, the lower limit of the ratio (8.5), as <5(Z)->0, exceeds at each point x except at most at those of E. Therefore, e being any positive number, this limit is non-negative for almost all points x. Combining this with the symmetrical result for the upper limit of the same ratio, we complete the proof. Another generalization of Theorem 6.1, also due to Ward, uses the following definition of relative derivation, which is slightly different from that given in Chap. IV, 2 (cf. A. J. Ward [3] and A. Eoussel [1]). Given two finite functions of a real variable F and G, we shall say that a number a is the Roussel derivative of the function F with respect to G at a point # , if when I denotes any interval containing # , we have (i) F(I) a-G(I)->0 and (ii) \F(I) a-<3(Z)|/O((?;Z)->0, as c5(/)->0 (the ratio in (ii) is to be interpreted to mean whenever its numerator and denominator vanish together; O ((?;!) denotes, in accordance with Chap. Ill, p. 60, the oscillation of G on I). [81 The Perron-Stieltjes integral. 211 When the oscillation of the function O at # is finite, the condition (ii) evidently implies (i); however, when 0(6?; a? ) = -foo, the condition (i) plays an essential part, whereas (ii) is then satisfied independently of F and of a. It is also to be observed that when o(G; x )/ -foo, and when F is a function which has the relative derivative .F^(a5 ) (cf. Chap. IV, 2, p. 109), the latter is also the Roussel derivative of F with respect to G. Finally, in the case of deri- vation with respect to monotone functions, the two methods are completely equivalent. In particular therefore, when G(x)^x, Roussel derivation with respect to G it equivalent to ordinary derivation. The proof of the theorem on Roussel derivability of the in- definite ^-integral is much the same as that of Theorems 6.1 and 8.4; it depends, however, on the following lemma which may be regarded as a generalization of a result of W. Sierpiriski [4]. (8.6) Lemma. Let O be a finite function of a real variable, E a bounded set in JK^ and 3 system of intervals such that each point of E is a (right- or left-hand) end-point of an interval (3) of arbitrarily small length. Then, given any number //<|G[]|, we can select 'from 3 o- finite system {!*) of non-overlapping intervals such that Proof. Suppose, for simplicity, that the set E lies in the open interval (0,1). For each positive integer n, let A n and B n denote respectively the sets of the points of E each of which is respectively a left- or right-hand end-point of an interval (3) contained in (0,1) and of length exceeding l/n. We evidently have E=lim(A n +B n ) and there therefore exists a positive integer n such that |G[^4n () + J Suppose, for definiteness, that |G[A/i ]|> ^//. Now, it is easily seen that, if \G[A no ]\=+oo, there exists a point x such that \O[A n(t - J"]|=-|-oo for any interval*/ containing X Q in its in- terior. Hence, from the family of intervals (3) whose left-hand end- points belong to A no and whose lengths exceed l/Ji , we can ob- viously select an interval / so as to have \G[I]\ ^ |(?[A V /]|> .V/w. Suppose now that \G[AnJ\<+oo. Then, by induction, we can extract from 3 a finite sequence of intervals {I*=(a*, b*)}*=i,2 ..... P in such a manner that, writing for symmetry ft =0 and a p +i=l, we have: (i) b k a k > l/n for fc=l, 2, ..., p, (ii) b k -\ < a k and for fc=l,2,...,p, and (iii) the 212 CHAPTER VI. Major and minor functions. interval (b pj a p j t .\) = (b p , 1) contains no points of A no . Since, on account of (i), we certainly have p < n , it follows from (ii) and (iii) that Z\G[I*]\^\G[A n J\-p-(\G[A n J\~-]iti)/n >} [ ti, i. e. that k the system of intervals {Ik} fulfils the required conditions. (8.7) Theorem. Every finite function f which is &$-integrable with respect to a function G on an interval JT , is the Roussel derivative with respect to G of its indefinite ^-integral at each point x of I Q except at most those of a set E such that \G[E]\ = Q. Proof. Let e be any positive number and U a major function of / with respect to G such that J7(/ ) -P(/ ) <* 2 , where P de- notes the indefinite ^-integral of /. Let us write H=U P, and denote by E e the set of the points x of I for which there exist intervals I of arbitrarily small lengths, such that x e I and that H(I)^t-\G[I]\. It follows that each point of E is an end-point of intervals /, as small as we please, which fulfil the inequality -\G[I]\. Therefore, denoting by /* any number le&s than and applying Lemma 8.6, we can determine in I a finite system of non-overlapping intervals </*} such that H (I*)^!e-|6?[I*]| for fc=l, 2, ..., p and that J*| #[/*]! ^ I/'. Consequently, since H k is non-decreasing, 2 >#(/ )^u/4; and therefore ^<4e. and hence Now let x be any point of I . We have for every sufficiently small interval I containing #, )-f(x) G(I)=U(I)-f(x) G(I) and, unless x belongs to the set JE? 8 , we also have Combining this with the similar upper evaluations of P(I) f(x)G(I) obtained by symmetry, we see, since 6 is an. arbitrary positive number, that / is the Eoussel derivative of the function P with respect to G, at every point x of I except at most those of a set E such that \G[E]\ = 0. CHAPTEE VII. Functions of generalized bounded variation. 1. Introduction. The definition adopted in Chap. I ( 10) as starting point of our exposition of the Lebesgue integral, con- nects the latter with the conception of definite integral due to Leibniz, Cauchy and Riemann (cf. Chap. I, 1 and Chap. VI, 1). On ac- count of the results of 7, Chap. IV, we may, however, also regard the Lebesgue integral as a special modification of that of Newton (cf. Chap. VI, I) and define it as follows: (L) A function of a real variable f is integrable if there exists a function F such that (i) F'(x) f(x) at almost all points x and (ii) F is absolutely continuous. The function F (then uniquely determined apart from an ad- ditive constant) is the indefinite integral of the function f. A definition of integral is usually called descriptive when it is based on differential properties of the indefinite integral and therefore connected with the Newtonian notion of primitive; this is the case of the definition (L) of the Lebesgue integral. In the note of F. Riesz [9] the reader will find an elementary and elegant account of the fundamental properties of the Lebesgue integral based on a descriptive definition differing slightly from the one given above (an ac- count based directly' on the definition (L) is given in the first edition of this book). By contrast to the descriptive definitions, we call constructive the definitions of integral which are based on the conception of definite integral of Leibniz-Cauchy, i. e. on approximation by the usual finite sums. Thus for instance, the classical definition given by H. Lebesgue [1] in his Thesis may be regarded as constructive (the reader will find a very suggestive explanation of this definition in the note by H. Lebesgue [8]); cf. also the definitions of Lebesgue integral given in the following memoirs: W. H. Young [3], T. H. Hil- debrandt [1], F. Riesz [1] and A. Denjoy [7; 8], As is readily seen, the definition (L) constitutes a modification of that of the integral of Newton, in two directions: firstly, a gen- eralization which enables us to disregard sets of measure zero 214 CHAPTER VII. Functions of generalized bounded variation. in the fundamental relation F'(x) = f(x); and secondly, an essential restriction, which excludes all but the absolutely continuous functions from the domain of continuous primitive functions con- sidered. Some such restriction is, in fact, indispensable, unless we give up the principle of unicity for the integral: to see this it is enough to consider, for instance, singular functions which are continuous and not constant, and whose derivatives vanish almost everywhere (cf. Chap, in, 13, p. 101). But although the condition (ii) cannot be wholly removed from the definition (L), it is possible to replace it by much weaker conditions, and the corresponding generalizations of the notion of absolute continuity give rise to extensions of the Lebesgue integral, known as the integrals 3^ and of Denjoy. We shall treat in this Chapter two generalizations of absolutely continuous functions: the functions which are generalized abso- lutely continuous in the restricted sense or ACG*, and those which are generalized absolutely continuous in the wide sense or ACG. If, in the definition (i), we replace the con- dition (ii) by the conditions that the function F is ACG* or ACG respectively,, we obtain the descriptive definitions of the integrals ff+ and >. It must be added however that the second <rf these defini- tions requires a simultaneous generalization of the nation of deriv- ative, to which is assigned the name of approximate derivative (or asymptotic derivative) and which corresponds to approximate continuity (vide Chap. IV, 10). A function which ia ACG (unlike those which are absolutely continuous or which are ACG*) may in fact fail, at each point of a set of positive measure, to be derivable in the ordinary sense, and yet be almost everywhere derivable in the approximate sense. Therefore, in order to obtain the definition of the integral & from the definition (), it is necessary not only to modify the condition (ii) as explained above, but also to replace in the condition (i) the ordinary by the approximate derivative. The integrals 2>* and S> will be studied in the next chapter; the preliminary discussion of their definitions just given, is intended to emphasize the important part played by the generalizations of the notion of absolute continuity, which are treated in this chapter. The results of which an account is given in the following are essentially due to Denjoy, Lusin and Khintchine. The first definition of the integral $+ was given in notes dating from 1912 by A. Denjoy [2; 3] who employed the constructive method based on a transfinite process (vide Chap. VIII, 5). These notes at once attracted the attention of N. Lusin [2] who originated the descriptive theory of this integral. Finally, A. Khintchine [1; 2] and [2] A theorem of Lusin. 215 A. Den joy [4] defined, independently and almost at the same time, the process of integration ^ as a generalization of the integral ^*. A systematic account of these researches may be found in the memoir of A. Den joy [6]. As shown by W. H. Young [6] the generalization of the Denjoy inte- grals can be carried still further if we give up, partially at least, the continuity of the indefinite integral. For subsequent researches in this direction, vide J. <\ B ur- kill [5; 6; 7], J. Ridder[6;7], M. D. Kennedy and S. Pollard [-1], S. Ver- blunskyfl], and J. Marcin ki e wicz and A. Zygmund [1]. Except in a few general definitions in 3, we shall consider in this chapter only functions of a real variable. As therefore we shall be employing in B notions established in the preceding chapters for arbitrary spaces R m , it will be convenient to add a few com- plementary definitions. We shall say that a point a is a right-hand point of accumulation for a linear set E, if each interval [a, a + h], where &>0, contains an infinity of points of E. A point of E which is not a right-hand point of accumulation for the set E is termed isolated on the right of this set. The definitions of left-hand points of accumulation and of points isolated on the left are obtained by symmetry. Similarly, for each linear set E, in addition to the densities defined in 10, Chap. IV, we define at each point x four unilateral densities: two outer right-hand, upper and lower, and two outer left- hand, upper and lower, densities of E. We shall understand by these four numbers the values of four corresponding Dini derivates of the measure-function (cf. Chap. IV, 6) of E at the point x. If at a point x, two of these densities on the same side (right or left) are equal to 1, the point x is termed unilateral fright- or left-hand) point of outer density for the set E. The term "o liter" is omitted from these expressions if the set E is measurable. Finally, we shall extend the notation of linear interval and denote, for each point a of JS 1? by ( oo, a), (00, a], (a, +) and [a, +00) the half-lines x<a, x^a, x >a and x~^a respectively. * 2. A theorem of Lusln. While discussing the significance of the condition (ii) in the definition (L) of an integral, we remarked that a continuous function which is almost everywhere derivable is by no means determined (apart from the additive constant) when we are given its derivative almost everywhere. It is, however, of greater interest that, for a function /, the property of being almost everywhere the derivative of a continuous function, itself represents no restriction at all, except, of course, in so far as it implies that 216 CHAPTER VII. Functions of generalized bounded variation. the function / is measurable and almost everywhere finite (this last assertion follows, for instance, from the corollaries to Theorem 10.1, p. 236). We shall prove this result, which is due to N. Lusin [I; 4] (cf. also B. W. Hobson [II, p. 284]), by means of two lemmas. (2.1) Lemma. If g is a function summdble on an interval [a, ft], there exists, for each >0, a continuous function G such that (i) G f (x)=g(x) almost everywhere on [a, ft], (ii) Q(a) G(b)=Qj and (iii) |G(0)|^e at every point x of [a, 6]. Proof. Let H(x) be the indefinite integral of g(x). We insert in [a, ft] a finite sequence of points a =oo<ai <...< ft such that the oscillation of H is less than e on each of the intervals [a/, a/4.,] where i=0, 1, ..., n 1. Let F (cf. (13.4), Chap. Ill, p. 101) be a function which is continuous and singular on [a, ft], monotone on each interval [a/, a/+i] and coincides with the function H at the end-points of these intervals. Writing G=HFj we shall have (i) Q'(x)=H'(x)F'(x)=H'(x)=g(x) at almost all the points x of [a, ft], (ii)0(a) = ff(ft)=0, and finally (m)\0(x)\=\H(x)F(x)\^e on each interval [a/, a/+i], and therefore on the whole interval [a, 6]. (2.2) Lemma. If g is a function which is summable on an interval J=[a, 6] and if P is a closed set in J, there exists for each > a con- tinuous junction G such that (i) G f (x) = g(x) at almost all the points x of JP, (ii) G(x)=Q and G'(x)=Q at all the points x of P and (iii) |G(a?+fe)|^e.|ft| for every x'of P and every h. Proof. Let us represent the open set JP as the sum of a se- quence {/A=(a*,ft*)}*=i,2,... of non-overlapping open intervals, and in- sert in each interval I k an increasing sequence of points {a^} /== <t _ 1|0 lt ... infinite in both directions and tending to a* or ft* according as -> oo or i-+oo. Let us further denote, for each fc=l, 2, ..., and i=0, 1, 2,..., by <? the smaller of the numbers f -(ag> a A )/(*+|t|) and e-(b k a ( *+ l) )/(k + \i\). Lemma 2.1 enables us to determine in each open interval I k a continuous function G* such that Gk(x)=g(x) almost everywhere on I A , G(ajf>)=0 for i=0, 1, 2, ..., and ifl^aOKcjP when a( ?^ x <<%+ l) - H we now write G(x)=G k (x) for xelk and 4=1, 2,..., and G(x)= elsewhere on R^ we see at once that the function G is continuous and fulfils the required conditions (i), (ii) and (iii). [2] A theorem of Lusin. 217 (2.3) Lusin 9 8 Theorem. If f is a function which is measurable and almost everywhere finite on an interval J=[a, b], there always exists a continuous function F such that F'(x) = f(x) almost everywhere on J. Proof. We shall define by induction a sequence of continuous functions {G n } n =Q,i t ...) each of these functions being almost everywhere derivable, and a sequence of closed sets {P/,)/i-o,i, .. in J, such that, /i /i writing Q n =Ph and F n Gk, the following conditions will be *=o k^ o satisfied for n= 1, 2, .... (a) F n (x) = f(x) for xeQ n , (b) O n (x) = for xeQ n __ l9 (c) \G n (x-\-h)\^\h\/2" for every xeQ n ^ and every h, (d) |j_gj<i/ n . For this purpose, we choose G (x) = Q identically and P 0, and we suppose that for n~ 0, 1, ...,r the closed sets P n and the continuous functions G n , almost everywhere derivable, have been defined so as to satisfy the conditions (a), (b), (c) and (d) for each n^r. Since the function / is measurable and almost everywhere finite, and since the function F r is almost everywhere derivable, we can determine a measurable subset E r of J~Q r such that (2.4) \j^Q r ^.E r \ and such that the derivative F' r (x) exists at each point x of the set E r and is bounded, together with the function /(a?), on this set. Hence by Lemma 2.2, we can determine a continuous function (? r -H, almost everywhere derivable, in such a manner that (i) G' r +\(x) = =f(x)F' r (x) at almost all points of E r J Q r , (u)O r -i-\(x)=G f r+ \(x)=0 at all points of Q n and (iii) \G r +i(x + h)\^\h\l2 r + l for every xeQ r and every h. Now it follows from the first of these conditions and from (2.4), that there exists a closed set P^iC^r such that: (2.5) \J-Q-P r+l \<\l(r+l), (2.6) G' r ^(x)=j(x)-F' f (x) for and we easily verify, on account of (2.6), (ii), (iii) and (2.5), that the conditions (a), (b), (c) and (d), still remain valid for n = r+l. Let us now write: (2.7) F(x) = }imF*(x)=:G k (x), (2.8) Q^ 218 CHAPTER VII. Functions of generalized bounded variation. In view of the condition (c), the series occurring in (2.7) con- verges uniformly,, and the function F is therefore continuous. Let # be any point of Q. Then for every sufficiently large integer n we have XQ + QH, and since h h ' ^J h we find, on account of the conditions (a), (b) and (c), that F(a lim sup - and so r that F'(x Q ) = f(x ). Now it follows from the condition (d) that |J $| = 0; we therefore have F'(x) = f(x) at almost all the paints x of J, and this completes the proof. Theorem 2.3 remains valid for any space K m : If f is a measurable function which is almost everywhere finite in a space R m , iktre exists an additive continuous function of an interval F sMch that F'(x) f(x) almost everywhere in R m - The proof is almost the same as that ol THie<*rn 2.3. We may also, in the foregoing statement, replace the ordinary derivative F*(x) by the strong de- rivative (vide Chap. IV, 2, p. 106), but the proof is then more elaborate. It may be remarked further that Lusin's theorem in the form (2.3), is obvious if the function / is summable; for / is then almost everywhere the derivative of its indefinite integral. But this is no longer BO when we wish to determine a function F with a strong derivative almost everywhere equal to / (cf. Chap. IV, p. 132). Nevertheless, it can be shown that given in a space Km any summable function of a point /, there always exists an additive continuous function of an inter- val, of bounded variation, F, such thai F f s (x) f(x) almost everywhere in jRm. Lusin's method is applicable in several other arguments. It has been used, for instance, by J. Marcinkiewicz [1], to derive the theorem: There exists a continuous function of a real variable F which has the following property: with each measurable function /, almost everywhere finite, there can be associated a sequence of positive numbers {h n } tending to such that lim [F(x+h n )F(x)]lhn=f(x) n at almost all the points x. $ 3. Approximate limits and derivatives. Given any function F defined in the neighbourhood of a point X Q of a space R m , we shall call approximate upper limit of F at x the lower bound of all the numbers y (+00 included) for which the set E[F(x)>y] JC has x as a point of dispersion (cf . Chap. IV, 10). Similarly, the approximate lower limit of the function F at the point x is the [3] Approximate limits and derivatives. 219 upper bound of the numbers y for which the set E(\F(#)<t/] has x X as a point of dispersion. These two approximate limits of F at X Q are called also extreme approximate limits and denoted by lim sup &ipF(x) X->XQ and lim inf ap F(x) respectively. When they are equal, their com- mon value is termed approximate limit of F at X Q and denoted by lim a It is easily seen that if E is * measurable set for which x is a point of density, then, in the preceding definitions of extreme approximate limits, the sets fj[F(x)>y] and ]&[F(x)<iy] may be replaced by the sets E[F(x) >y; xe E] and JT x respectively. Hence (3.1) Theorem. If two functions coincide on a measurable set E, their approximate extreme limits coincide at almost all points of E, and in fact at every point of density of E. We see further that if x is a point of density for a, measurable set E and if the limit of F(x) exists as x tends to X Q on E, then this limit is at the same time the approximate limit of F at the point a? . Therefore, if a function F is approximately continuous (ef. Chap. IV, p. 131) at a point x^ we must have F(# ) = lim ap F (x). J^*o If X Q is a point of density for a measurable set E and if, further, the function F is measurable on E, it i easily seen that the ap- proximate upper limit of F at x is the lower bound of the num- bers y for which the set E[.F(a?)^y; #e E] has x % as a point of density. It follows, by the definition of approximate lower limit, that with the same hypotheses on the set E and on the function F, in order that l = lim ap F(x), it is necessary and sufficient that for each e>0 the set E[Z f *^F(x)^l+; xcE] should have the point X Q as X a point of density. Let us remark finally that the following inequalities hold be- tween approximate and ordinary extreme limits: (3.2) and hence the approximate limit exists and is equal to the ordinary limit, wherever the latter exists. 220 CHAPTER VII. Functions of generalized bounded variation. In order to understand better the meaning of the definitions of ap- proximate limits, it may be remarked that the definitions of the ordinary limits are expressible in a very similar form. Thus the upper limit of F(x) at x may be defined as the lower bound of all the numbers y for which x is not a point of accumulation for the set E [F(x)>y]. The inequality (3.2) then becomes obvious. JC For functions of a real variable, in addition to the approximate limits defined above, and which in this case we call bilateral, we introduce also four unilateral approximate limits. The approximate upper right-hand limit of a function F at a point x is the lower bound of the numbers y for which the set E[F(x)>y; #># ] has # as or a point of dispersion. This limit is written lim sup ap F(x). The Jr->.r -f three other approximate extreme unilateral limits are defined and denoted similarly. These generalizations of the notion of limit lead very naturally to parallel generalizations of derivates. Thus, given a finite func- tion of a real variable F, we define at each point x the approximate right-hand upper derivate F^(x ) and lower derivate F P (XQ), the approximate left- hand upper derivate F^(x ) and lower derivate F-AP(XQ), and the approximate bilateral upper derivate F^(x Q ) and lower derivate ^ 7 a P (.r ), as the corresponding approximate extreme limits of the rat ; o [F(x)~ F(x Q )]l(x X Q ) as x-+x . When all these derivates are equal (or, what comes to the same, when ^ ap (iP )=^ T ap(iP )), their common value is called approximate derivative of F at X Q and is denoted by jFa p (# ); ^ further, this derivative is finite, the function F is said to be approximately derivable at X Q . For some further generalizations, such as "preponderant derivates" ("nombres d^riv^s pr^ponderants"), and for a deeper study of the prop- erties of approximate derivates, the reader should consult A. Den joy [6] and A. Khintchine [5]. The properties of bilateral approximate limits, discussed above, can be taken over, with the obvious formal modifications, so as to apply to unilateral approximate limits. In particular, Theorem 3.1 may be completed as follows: (3.3) Theorem, ff two functions of a real variable coincide on a mea- surable set E, their approximate extreme limits and their approximate derivates coincide respectively at almost all points of E, and in fact at every point of density of E. Also, if a function F is measurable on a set E, we have F- dp (x)=Fs(x) at almost all the points x of E at which the function F has a derivative with respect to the set E. I 4] Functions VB and VBG. 221 $ 4. Functions VB and VBG. We shall denote by V(jF; J0), and call weak variation of a finite function F(x) on a set E, the upper bound of the numbers 2\F(bi) -P(a/)| where {[a/, fr/]} is any se- / quence of non-overlapping intervals whose end-points belong to E. If V(FjE) <+oo, the function F is said to be of bounded variation in the wide sense on the set E, or, simply, of bounded variation on J5, or VB on E. In the special case in which the set E is a closed interval, we clearly have V(F; E) W(F; E), i. e. the weak variation of the function F on E then coincides with its absolute variation in the sense of Chap. Ill, 13. The definition of functions of bounded variation in the wide sense on a set thus constitutes a generalization (for functions of a real variable) of that of func- tions of bounded variation on an interval. If 7 is a linear figure formed of disconnected intervals we only get the inequality V(F- 9 E)^'W(F;E), but it is easy to see that even then the relation W(F;E)<i-\-co always implies V(F',E)<+oo. Plainly, every function which is VB on a set E is bounded on E and is VB on each subset of E. Again, any function F which is continuous on a set E and VB on a set A C E everywhere dense in E (cf. Chap. II, 2) is VB on the whole set E (for then V(jP; .E) = V(jP; A)). Finally, if F and G are two functions which are bounded on a set E and M denotes the upper bound 6f the absolute values of these functions on E, we have V(aF + bG-,E)^\a\-V(F-,E) + \b\-V(Q-,E) for each pair of constants a and ft, and V(JF-0; E)^M.[V(F; E)+V(G;E)] (cf. Chap. Ill, p. 97). Hence every linear combination^ with constant coefficients, of two functions which are VB on a set, and the pro- duct of the two functions, are themselves VB on this set. A function F(x) is said to be of generalized bounded variation in the wide sense on a set E, or simply, of generalized bounded variation on Ej or again, for short, VBG on JS?, if E is the sum of a finite or enumerable sequence of sets on each of which F(x) is VB. Prom what has just been proved for functions which are VB we see at once that every line.ar combination of two functions which are VBG on a set, and the product of the two functions, are themselves VBG on this set. (4.1) Lemma. In order that a function F be bounded and non- decreasing [of bounded variation] on a set E, it is necessary and sufficient that F coincide on E with a function which is bounded and non-decreasing [of bounded variation] on the whole straight line JR r 222 CHAPTER VII. Functions of generalized bounded variation. Proof. Let us denote for each #, by E( X) the set of the points of E which belong to the interval (- oo, #]. We shall consider two cases separately. 1 The function F is bounded and non-decreasing on E. For each x, let 0(x) denote the upper bound of the function F on the set E (x ^ or else the lower bound of the function F on JE, according as E (X )^=Q or E( X ) = Q. The function thus defined is evid- ently bounded and non-decreasing on the whole straight line 1^ and coincides with the function F on E. 2 The function F is VB on E. For each point #, let V(x) = y(F' J E (x} ) if E (X )=0, and V(x) = if E (x} = 0. We see at once that the function V(x) is monotone and bounded on the whole straight line JSj and that V(x) F(x) is non-decreasing and bounded on E. Hence, by what has just been proved in 1, there exists a func- tion 0(x) which is bounded and non-decreasing on JJj and which coincides on E with V(x) F(x). We have therefore F(x) = V(x) G(x) for every xeE, and since the function V(x) G(x), as difference of two bounded monotone functions, is clearly of bounded variation on jR 1? this completes the proof, (4.2) Theorem. Let F be a function which is measurable on a set E and which is VB on a set E^C_E. Then (i) F is approximately deriv- able at almost all points of E and (ii) there exists a measurable set E 2 such that J^C^aC^ <d thai F is VB on E 2 . Proof. By Lemma 4.1, there exists a function O which coin- cides with F on the set E l and which is of bounded variation on the whole straight line J^. Let E 2 be the set of the points x of E at which F(x) = 0(x). Then since F is, by hypothesis, measurable on E, the set E 2 must be measurable. Moreover, as J^C^iC-Ei the function F is, with <?, of bounded variation on E 2 , and by Lebesgue's Theorem 5.4, Chap. IV, and Theorem 3.3, the finite approximate derivative F* v (x)=G'(x) exists at almost all the points x of E 2 . Theorem 4.2 leads at once to the following theorem, which for the Den joy integral takes the place of Lebesgue's Theorem on derivability of functions of bounded variation: (4.3) Theorem of I^njo^-Khint chine. A function which i* measurable and VBG on a **t i* approximately derivable at almost all points of this set. [5] Functions AC and ACG. 223 Finally, if we make use of Theorem 9.1, Chap. IV, and Lemma 4.1, we may complete Theorem 4.2 as follows: (4.4) Theorem. A function F which is VB on a set E, is derivable with respect to the set E at almost all points of E. Moreover, if N de- notes the set of the points at which the derivative F'E(X) (finite or in- finite) does not exist, then the graph of the function F on N is of length zero and consequently the set of the values taken by F on N is of meas- ure zero-,. in symbols A(B(F;N)}\F[N]\ = Q. For an extension of Theorem 4.3 to functions of two variables, vide V. G. Celidze [1]. 5. Functions AC and ACG. A finite function F will be termed absolutely continuous in the wide sense on a set E, or absolutely continuous on E, or simply AC on E, if given any e>0 there exists an 77 >0 such that for every sequence of non-overlapping intervals {[*, b k ]} whose end-points belong to E, the inequality 5}(b k ak) < 77 implies 2\F(b k )F(a k )\<e. * A function F will be termed generalized absolutely continuous function in the wide sense on a set E, or generalized absolutely con- tinuous function on E, or finally ACG on E, if F is continuous on E and if E is the sum of a finite or enumerable sequence of sets E n on each of which F is AC. These definitions generalize that of functions absolutely con- tinuous on a linear interval (cf. Chap. Ill, 12, 13) and allow us to generalize certain fundamental properties of the latter. We see at once, by the arguments of the preceding , that every linear combina- tion of two functions which are AC [ACG] on a bounded set, and the product of such functions, are themselves AC [ACG] on this set. Further, every function which is AC on a bounded set E is VB on E. In fact, if F is such a function, there exists an rj Q >0 such that V(jP; E-I)^1 for each interval I of length < >?<,. It follows that F is bounded on E. Let M be the upper bound of the absolute values of F on E, and let J be an interval containing E\ then, J is the sum of a finite num- ber of non-overlapping intervals JuJv-iJp each of which is of length <7] , and we find V(J?; E)^%V(F; E>J k ) + 2pM<+co. It follows at once that any function which is ACG on a set E (bounded or unbounded) is VBG on E, and therefore, by the theorem of Denjoy-Khintchine given in the preceding , every function which is ACG on a mecksurable set is approximately derivable at almost all points of this set. 224 CHAPTER VII. Functions of generalized bounded variation. Nevertheless we can construct an example of a function which is ACG on an interval and which is not derivable in the ordinary sense at the points of a set of positive measure. For ' this purpose, let H denote a bounded, perfect, non-dense st of positive measure, with the bounds a and b. Let 1= [a, b] and let (I n = [a n , b n ]} be the sequence of the intervals contiguous to H. We denote further by p n the length of the largest subinterval of [a, b] which does not overlap the first n in- tervals I t , 7 2 , ... , / n of this sequence. Plainly (5.1) lim|IJ = and lira =(). n n Now let c n denote for each n=l, 2, ... , the centre of the interval I n , and let F be the function defined on the interval I by the following conditions: 1 F(x)=0 for xe H; 2 F(c n )=:\I n \+e n for ra=l,2, ...; 3 the function F is linear in each of the intervals [a n , c n ] and [c n , b n ] where n=l, 2, ... . Thus defined, the function F is continuous on I by (5.1) and is AC on H and on each I n ; since l n , it follows that F is ACG on I. We shall show that F is not derivable at any point x e E . In fact, since F vanishes on H, we have (5.2) (z)< Q^F(x) for every xeH. If therefore a point X Q is a left-hand end-point of an I n , there can be no derivative F'(x ) since it is clear that F (x ) = F~*~(x ) > and therefore, by (5.2), that F(x Q )^ F(XQ). Similarly, F(x ) < 0<-F(x ) if x is a right-hand end-point of an interval I . If, on the other hand, x H, x ^a n and x^b n for n=l,2,..., denote by i n the suffix of that interval of the system Ij, I 2 , ..., I n which is nearest to x . Then lim ! +00 and ^ \c in x \ < |// n |-r-^ n . and so, by the definition of F(x), we have n F(ei n )F(xQ)-=\It n \+etn : ^\Iin\+en > K| ^ol- since h c i n =x o> Jt f Nws that either n or -?(a? ) ^ 1, which by (5.2), proves that F is not dervable at x . us remark, in conclusion, that a function F which i* con- tinuous on a set E and which is AC on a subset of E everywhere dense in E, is AC on the whole set E. 6. Lusln's condition (N). A finite function F is said to fulfil the condition (N) on a set E, if \F[H]\ = Q for every set HTE of measure zero (for the notation cf. Chap. Ill, p. 100). Clearly, a function which fulfils the condition (N) on- each of the sets of a finite or enumerable sequence, also fulfils this condition on the sum of these sets. The condition (N) was introduced by N. Lusin [I, p. 109], who was the first to recognize the importance of this condition in the theory of the integral. It is easy to see that in the domain of continuous functions the con- dition (N) is necessary and sufficient in order that the function should transform every measurable set into a measurable set (cf. H. Rademacher[l] and H. Hahn [I, p. 586]). Among the more recent researches devoted to the condition (N) and to other similar conditions (cf., below, Chap. IX) the reader should consult above all N. Bary [3]. [6] Lusin's condition (N). 225 (6.1) Theorem* A junction which is ACG on a set necessarily fulfils the condition (N) on this set. Proof. Since each set on which a function is ACG is the sum of a sequence of sets on which the function is AC, it will suffice to prove that \F[H]\ = Q whenever H is a set of measure zero and F a function AC on H. For this purpose, let e be any positive number. We denote, for brevity, by M (E) and m(E) respectively the upper and lower bounds of F on E, when E is any subset of T, and we write M(E)=m(E) = Q in the case in which E~Q. Since the function F is AC on Hj there exists a number ??>0suchthat 2[M(H'I/ t )m(H-I fl )]<e k for every sequence of non-overlapping intervals {Ik} which sat- isfies the condition J?|/A| < rj- Now since the set H is of measure zero, we can determine a sequence of non-overlapping intervals {Ik} which satisfies this last condition and which covers, at the same time, the whole set E. Therefore, since |jP[ff-lA]|<i(ff-// r )~m(fl r -/^) for each ifc, it follows that |jP[If]|^. Hence, e being arbitrary, It follows from Theorem 7.8 (1), Chap. IV, that every func- tion which is absolutely continuous on an interval and whose deriv- ative is almost everywhere non-negative, is monotone non-decreasing. With the help of Theorem 6.1, this result can be extended to functions which are ACG and we have: (6.2) Theorem. Every junction F(x) which is ACG on on interval I and for which we have almost everywhere in this interval jPap(,r)^0, or more generally, F^(x) ^ 0, is monotone non-decreasing. In particular therefore y if the approximate derivative of a func- tion which is ACG on an interval vanishes almost everywhere on this interval, then the function is a constant. Proof. Let e be any positive number and let The function G is then ACG on the interval I (together with the function F), and moreover, we have G~*~(x)=F^(x) + e^e>0 at almost all the points x of I. Hence, denoting by H the set of the points x at which G^(x)^Q, we have |#| = 0, and this implies, by Theorem 6.1, that \O[H]\ = Q. Thus the set O[H] cannot contain any non-degenerate interval, and by Theorem 7.1, Chap. VI, the function G(x)~F(x) + ex is non-decreasing on I. It follows at once, by making e->0, that the function F is itself non-decreiusing. 226 CHAPTER VII. Functions of generalized bounded variation. If we analyze the preceding proof, we notice that the hypothesis of generalized absolute continuity of F(x) has been used only to show that every function of the form F(x)+(x, where e> 0, fulfils the condition (N). It is remarkable that the condition (N) need not remain satisfied when we add a linear function to a function fulfilling the condition, even when this last function is restricted to be continuous (vide S. Mazurkiewicz [1]). For this reason it is not enough to suppose in the preceding proof that the function F(x) merely fulfils the condition (N). Nevertheless, Theorem 6.2 itself does remain true for arbitrary functions which fulfil the condition (N). The theorems which will be proved in Chap. IX, 7, include a more general result, namely that every continuous function which fulfils the condition (N) and whose derivative is non -negative at almost all the points at which it exists, is monotone non -decreasing. We shall show (vide, below, Theorem 6.8) that for continuous functions of generalized bounded variation on closed sets, the con- verse of Theorem 6.1 is true, i. e. that in this case the condition (N) is equivalent to generalized absolute continuity. Similarly, for continuous functions of bounded variation the condition (N) is equivalent to absolute continuity in the ordinary sense. We shall begin with a lemma which will also prove useful elsewhere. (6.3) Lemma. If, for a finite function F, the inequalities F^(x)^M and F (x)^ M, where M is a finite non-negative number, hold at each point x of a set D, then \F[D]\^ M-\D\. Proof. Let e be any positive number. Let D n denote for each positive integer n the set of the points x of D for which we have F(t)F(x)^(M + e)>\tx\ whenever \tx\^l/n. The sets D n evid- ently constitute an ascending sequence and we see easily that D = lim D n . n . . With each D n we can associate a sequence of intervals (r}k--\&... which covers D n and fulfils the condition (6.4) and in which, further, no I ( P has length greater than 1/n. By defini- tion of D m we therefore have, for every pair x^x 2 of points of D H -fi\ the inequality \F(x 2 )-F(x l )\^(M + e)-\x 2 -x 1 \^(M + ^)^ l} \, so that \F[D n -fi } ]\<(M + e)-\I ( k\. In view of the inequality (<U) it therefore follows that, for every n, and, by making first w->oo and then e->0, we derive |F( D]\ < [6] Lusin's condition (N). 227 (6.5) Theorem. If a function F is derivable at every point of a meas- urable set D, then (6.6) Proof. We may clearly assume that the set D is bounded. Given any >0, let D n denote, for each positive integer n, the set of the points xeD, at which (n l)e^\F'(x) <n-e. We then have, by the preceding lemma, /1--1 JX and hence, e being arbitrary, the inequality (6.6). The formula (6.6) remains true when we replace in it the derivative F'(x) by any Dini derivate, provided however that we restrict the latter to be finite in D. The proof then becomes rather more elaborate and requires certain general theorems on derivates which will be established later (vide Chap. IX, 4). (6.7) Theorem. In order that a function F(x) which is continuous and VB on a bounded closed set E, be AC on E, it is necessary wild sufficient that F(x) fulfil the condition (N) on this set. Proof. In view of Theorem 6.1, it remains to bo shown that the condition is sufficient. Suppose then that F fulfils the condition (N) on E. Let a and fc be the bounds of J5, and let G denote the function which coincides with F at the points of E and is linear in the intervals con- tiguous to E. The function G is evidently continuous and of bound- ed variation, and fulfils the condition (N) on the whole interval Given any subinterval / = [, b] of [a , 6 ], let us denote by D the set of the points of I, at which the function G is derivable, and write H=ID. Plainly |JI| = 0, and therefore also \G[H]\ = 0. On the other hand, since the interval with the end-points G (a) and G(b) is contained in <?[/], we have by Theorem 6.f> b |ff (6) -ff (a)| < |ff [D]| + |ff [ff J| - |ff [D]| < [\0'(x)\ ds. Since this inequality is valid for every subinterval /^[a, &J f Ooi*o] an(l since by Theorem 7.4, Chap. IV, the derivative G'(x) is summable on [a ,6 ], it follows that the function G is AC on [> , ft ], and therefore that F is AC on the set E, where F and G coincide. 228 CHAPTER VII. Functions of generalized bounded variation. It is easy to see that the same argument leads to a more general theorem: in order that a continuous function F which is continuous on an interval Z be ab- solutely continuous on this interval, it is necessary and sufficient that F fulfil the condition (N) on 7 and that its derivative exist almost everywhere on 7 and be sum- mable on 7 . This theorem will again be generalized in Chap. IX, 7. (6.8) Theorem. In order that a function F which is continuous a/nd VBG on a closed set E be ACG on E, it is necessary and sufficient that F fulfil the condition (N) on this set. Proof. In view of Theorem 6.1, we need only prove the con- dition (N) sufficient. Now, since F is VBG on the set E, this set is expressible as the sum of a sequence of bounded sets {E n } such that the function F is VB on each E n . By continuity of F on the closed set E, we may suppose (cf. 4, p. 221) that each set E n is closed. Since further F fulfils the condition (N) on E, it follows from Theorem 6.7 that the function F is AC on each E n , and there- fore ACG on E. 7. Functions VB* and VBG*. We shall denote by V*(jF;#) and term strong variation of a finite function F on a set E, the upper bound of the sums O(F',I k ) where {/*} is any sequence of * non-overlapping intervals whose end-points belong to E (in accord- ance with Chap. Ill, p. 60, O(F;I k ) denotes the oscillation of F on the interval Ik). If V*(.F, J5)< + oo, the function F will be said to be of bounded variation in the restricted sense on the set E, or VB* on E. Following the order of the definitions of 4, we shall say further that a finite function is of generalized bounded variation in the restricted sense, or simply, is VBG* on a set E, if E is the sum of a finite or enumerable sequence of sets on each of which the function is VB*. In the special case in which the set E is a closed interval, we clearly have Vt(F;E) = V(F;E) = W(F;E). It is easy to see that we always have V(F;E)< s Vt(F;E); so that every function which is VB* on a set, is VB on this set, and consequently, every function which is VBG* on a set, is VBG on this set. We next observe (by using trivial inequalities for the VB* case, and thence passing on to the VBG* case) that every linear combination, with constant coef- ficients, of two functions which are VB* [VBG*], and also the product of two such functions, are themselves VB* [VBG*]. [7] Functions VB* and VBG*. 229 Let us observe that, for a function, the property of being VB, VBG, AC, or ACG, on a set E depends solely on the behaviour of the function on E; whereas the property of being VB* or VBG* on E depends on the behaviour of the func- tion on the whole of an interval containing the set E. In other words, of two functions which coincide on a set E, one may be VB* or VBG* on E and the other not. The same remark applies to the property of being AC* or ACG* with which we shall be concerned in the next . We have remarked in 4, p. 221, that a function which is continuous on a set E and which is VB on an everywhere dense subset of E) is necessarily VB on E. A similar result is true for func- tions which are VB*, the assumption of continuity of the given function being now superfluous. We have in fact: (7.1) Theorem. Every finite function F which is VB* on a bounded set E is equally so on the closure E of this set. Proof. Let a and b denote the bounds of E and therefore also of E. Let a^=a <a 1 <...<a n =6 be any finite sequence of points of E; we write I=[a, b] and Z*=[aA-i, a*] for ifc=l,2, ... , n. We shall say that an interval Ik is of the first classifit contains points of Ej and otherwise of the second class. The intervals I I and I n are clearly of the first class, and we see easily that, if an interval Ik is of the second class, then both the adjacent intervals IA-I and I*+i are certainly of the first class. Let us denote by I = t <t 1 <...<t r = n the suffixes of the intervals I k of the first class and by ]Q<JI<.-.<JS those of the second. With each interval I i of the first class we associate a point h b h li -E and we write J/,= [&/,-!,&/,] for ft=l,2, ...,r. It is easy h to see that and n r Hence, O(F; I k ) ^ 3-ZO(F- t J h )+2-O(F' 1 IKS-HW; E)+O(F; I)], *=1 _ and therefore V,(.F; JJ )^3-[V t (F;E) + O(F ;I)]<+ 30 - This completes the proof. 230 CHAPTER VII. Functions of generalized bounded variation. (7.2) Theorem. If a function F is VBG* on a set E, then F is derivable at almost all points of this set] and further if N denotes the set of the points x of E at which the function has no derivative, finite or infinite, then \F[N]\ =A[E(F; JV)} = 0. Proof. We may clearly suppose that the set E is bounded and that the function F is VB* on E. Moreover, by Theorem 7.1, we may suppose that the set E is closed. Let therefore a and b denote the bounds of E on the left and on the right, and {I/,}/, -1,2,. . the sequence of the intervals contiguous to E. Writing m n and M n respectively for the lower and upper bounds of F on /, we define two functions m(x) and M(x) on [a,ft] making rn(x) m n and M(x)=M n for xeln where w=l, 2, ... , and m(x)= M(x) = F(x) for xeE. The two functions m(x) and M(x) thus defined are plainly of bounded variation on the whole interval [a, b] and coincide with F(x) on the set E. Therefore, denoting by N the set of the points x e E at which either one at least of the (finite or infinite) derivatives M' (x) and m'(x) does not exist, or both exist without being equal, we find by Theorem 9.1, Chap. IV, that (7.3) |F[jyH4B(JF;tf ))| = 0. On the other hand, m(x)=F(x) M (x) at every point x of E, while m(x)^F(x)^M (x) on the whole interval [a, ft]. It follows that the derivative F'(x) = m'(x)=M f (x) exists at each point x of U, except at most those of the set N which is subject to the rela- tion (7.3). Finally, since the functions m(x) and M(x) are deriv- able almost everywhere on the interval [a, ft], the function F must be derivable at almost all points of E, and this completes the proof. Theorem 7.2 (for continuous functions and in a slightly less complete form) was first proved by Denjoy and by Lusin, independently. It plays in the theory of the Denjoy -Perron integral (vide, below, Chap. VIII) a part similar to that of Lebesgue's Theorem (Chap. IV, 5) in the theory of the Lebesgue integral. A cor- responding part is played in the theory of the Denjoy-Khintchine integral by Theorem 4.3. But the latter is stated in terms of approximate derivation (cf. the example of p. 224) whereas Theorem 7.2, which requires no modification of the notion of derivative, is, for functions of a real variable, a direct generaliza- tion of Lebesgue's Theorem. [8] Functions AC* and ACG*. 231 8. Functions AC and ACG*. A finite function F is said to be absolutely continuous in the restricted sense on a bounded set E, or to be AC* on E, if F is bounded on an interval containing E and if to each e>0 there corresponds an ?y>0 such that, for every finite sequence of non-overlapping intervals {Ik} whose end-points belong to Ej the inequality ^|/*|<^ implies 2O(F;I k )<e. A function will be termed generalized absolutely continuous on a set Ej or ACG* on E, if the function is continuous on E and if the set E is expressible as the sum of a sequence of bounded sets on each of which the function is AC*. In the case in which the set E is an interval, the class of func- tions AC* on E coincides with that of the functions which are ab- solutely continuous on E in the ordinary sense. Every function which is AC* on an arbitrary set E is AC on E, and every function which is ACG* on E is ACG on E. On the other hand, any function which is AC* on a bounded set is VB^ on this set, and therefore, any function which is ACG* on a set is VBG* on this set. To see this, let F be AC* on a bounded set E. We can then determine a positive number r] such that V*(^;S-Z)^1 for every interval I of length less than r} . Let J be the smallest interval containing E, let M be the upper bound of \F(x)\ on J, and suppose J expressed as the sum of a finite number of non-overlapping intervals J 1? <7 2 , ... , J p each of length less than ry . We shall then have V*(F; E) ^ V+(F; E-J k ) + 2Mp ^ (2M+I).p < + oo, A 38 *! and this shows that the function F is VB* on E. Thus a function which is AC* on a bounded set E is both AC and VB* on this set, and similarly a function which is ACG* on E is both ACG and VBG* on E. The converse also is true, provided that the set E is restricted to be closed (vide, below, Theorem 8.8). Instead of giving a special proof of this result, we shall establish some more general theorems about the relations between the notions VB, AC, VB*, AC*, VBG, ACG, VBG* and ACG*. (8.1) Lemma. Let E denote a bounded closed set, {<//,} the sequence of the intervals contiguous to E, and I the smallest interval containing E. Then, for any function F which is finite on 7 , we have (8.2) 232 CHAPTER VII. Functions of generalized bounded variation. Proof. Let M , m and M , ra be the bounds (upper, lower) of Fj on E and on I respectively. Let MQ be any finite number less than M Q , and X Q a point of I such that M'o^F(x ). If we have # eJ5, this inequality implies M' Q ^M, while if # belongs to an interval, J ko say, of the sequence {</*!, jJfo<^ M+O(F; J k(> ). Hence (8.3) M ^ and similarly (8.4) m ^ On subtracting (8.4) from (8.3), we obtain, since M the relation (8.2). (8.5) Theorem. In order that a function F which is VB [AC] OTI a bounded closed set E, be VB* [AC*] on E, it is necessary and suf- ficient that the series of its oscillations on the intervals contiguous to E be convergent. Proof. The necessity of these conditions is obvious (cf. above p. 231); we have therefore only to prove them sufficient. Let then {J k \ denote the sequence of the intervals contiguous to E) and suppose that (8.6) We shall consider the two cases separately: 1 The function F is VB on E, i.e. V(F;E)< + oo. Then by Lemma 8.1, we have for every sequence {I n } of non-overlapping intervals whose end-points belong to jB, It follows by (8.6) that Vt(F;E)< + oo, i. e. that the function F is VB* on E. 2 The function F is AC on E. Then, given any e>0, there exists a number rj > such that, for every sequence of non- overlapping intervals {!} whose end-points belong to E, the in- equality 2\I n \<ri implies ^V(^; J5M n )<e/2. Now by (8.6), there n n exists a positive integer fc such that (8.7) Denote by rj the smallest of the fc +l numbers ??, | JJ, |JJ, ..., and let {/} be any sequence of non-overlapping intervals with end- Definitions of Denjoy-Lusin. 233 points in E, the sum of whose lengths is less than rj Q . None of these intervals / can contain one of the first fc intervals of the sequence {</}, and it follows from (8.7) and from Lemma 8.1, that Therefore the function F is AC, on E, and this completes the proof. (8.8) Theorem. In order that a junction F be AC, [ACG,] on a bound- ed closed set J5, it is necessary and sufficient that F be both VB, and AC [VBG, and ACG] on E. Proof. The necessity of these conditions is obvious, so that we have only to prove them sufficient. Now, if the function F is both VB, and AC on E, it follows at once from Theorem 8.5 that F is AC, on E. If on the other hand, F is VBG, and ACG on J5, we can express the set E as the sum of a sequence of sets {E n } on each of which F is both VB, and AC. Since F is ACG, and so continuous, on the set E, which is by hypo- thesis closed, F is AC on the closure E n of each E n . Similarly, by Theorem 7.1, F is VB, on each E n . Therefore by what has just been proved, F is AC, on each of the sets E n and so, ACG, on the set E. Theorem 8.8 ceases to hold if we remove the restriction that the set E is closed. Let E be the set of irrational points, and \ n [ n ^i i2 ,... the sequence of ra- tional points, of the interval [0, 1]; and let F(x)^=0 for x e E, and F(a n ) l/2 n for n=l,2, ... . The function F thus defined is evidently VB, and AC on E. To show that F is not AC,, nor even ACG,, on E, suppose that the set E is the sum of a sequence of sets {E n } on each of which F is AC,. By Baire's Theorem (Chap. II, Theorem 9.2), one at least of the sets E n would be everywhere dense in a (non- degenerate) subinterval of [0, 1], But this is plainly impossible, since every sub- interval of [0, 1] contains, in its interior, points of discontinuity of the function F. 9. Definitions of Denjoy-Lusin. The definitions which we have adopted in this chapter for the classes of functions VBG, ACG, VBG, and ACG, are based on the ideas of A. K hint chine [3]. Bather different definitions were given by N. Lusin [I] and A. Den joy [6], which are equivalent to those of Khintchine when we restrict ourselves to continuous functions. We give them here, in the form of necessary and sufficient conditions, in the following theorem. (9.1) Theorem. In order that a function which is continuous on a closed set J57, be VBG [VBG,, ACG, ACG,] on E, it is necessary and sufficient that every closed subset of E contain a portion on which the function is VB [VB,, AC, AC,]. 234 CHAPTER VII. Functions of generalised bounded variation. Proof. We shall deal only with the VBG ease, the proof for the other three cases being quite similar. 1 The condition is necessary. Let F be a function which is continuous and VBG on E. We can then express the set E as the sum of a sequence of sets {E n } on each of which the function F is VB and, by continuity of F, the sets E n may be supposed closed. Then by Baire's Theorem (Chap. II, 9), every closed subset of E' has a portion P contained wholly in one of the sets E n . The func- tion F) which is VB on each E nj is thus certainly VB on P. 2 The condition is sufficient. Suppose that F is a continuous function on E and that every closed subset of E contains a portion on which F is VB. Let {I n } be the sequence of all the open intervals / with rational end-points such that F is VBG on E-I. Let Q=^E-I n n and H=EQ. Plainly F is VBG on Q and we need only prove that the set H is empty. Suppose therefore that H 4- 0. Since H is clearly a closed set, there exists, by hypothesis, an open interval J such that H-J 4^0 and that the function F is VB on H-J. We may evidently assume that the end-points of J are rational. Therefore, the function F, which is VBG on the set Q, is also VBG on the set E-JCH-J+Q. This requires J to belong to the sequence of intervals {I n } and we have a contradiction, since the set JT, by definition, has no points in common with any of the intervals /. Theorem 9.1 shows in particular that every continuous function which is VBG on an interval I is at the same time VB on some subinterval of I. It fol- lows that for every continuous function which is VBG on an interval /, there exists an everywhere dense system of subintervals on each of which the function IB almost everywhere derivable, although this function may, as shown in 5, have no derivative at the points of a set of positive measure. 10. Criteria for the classes of functions VBG*, ACG, VBG and ACG. A series of theorems enabling us to distinguish certain types of functions of generalized bounded variation and cer- tain types of generalized absolutely continuous functions, are due to A. Denjoy [6], (10.1) Theorem. If F(x) is a function which fulfils at all points of a set, except at most those of an enumerable subset) one of the inequalities (10.2) F(x)< + oo or F(x)>oo J then the function F(x) is VBG* on this set. [ 10] Criteria for the classes of functions VBG* , ACG* , V BG and ACG. 235 Proof. It is enough to show that the set E of the points at which we have, say, F(x)<+oo, is the sum of an enumerable in- finity of sets on each of which F is VB*. For any positive integer n, let E n denote the set of the points x of E such that for every t, (10.3) 0<\t-x\^l/n implies [F(t)-F(x)]/(tx)^n. Further, for each integer i, let E' n denote the part of E n sit- uated in the interval [i/n, (t-fl)/n], and a n , b l n the lower and upper bounds of those of the E l n which are not empty. We have 00 OO 4 00 clearly E=^E,,=^ ^E' n . n 1 n 1 i oo Let now F n (x)=F(x)nx. For every point xcE n and for every point t which fulfils the first of the inequalities (10.3), we then have [F n (t)F n (x)]l(tx)^Q. In particular, given any pair of points x^x 2 (where x l ^x 2 ) of E' n9 we obtain (10.4) F n (a' n )^F n (x l ) ^ F n (x 2 ) and for every t such that #^^#2 we ^ n ^ that This last relation implies that, for every interval I=[a,j8] whose end-points belong to the set En, we have Q(F n iI)=F n (a)F n (P)j and therefore by (10.4), for every sequence {// =[;,#/]} of such intervals (which do not overlap), J j The function F n (x), and therefore also the function F(x)=F n (x) + nx, is thus VB^, on every set E l n and this completes the proof. (10.5) Theorem. If F(x) is a function which fulfils at all points of a set Ej except, perhaps, at those of an enumerable set, one at least of the conditions (10.6) oo<F+(x)^F+(x)<+oo or oo<F-(x)<F~(x)<+oo J then the set E is the sum of an at most enumerable infinity of sets on each of which the function F is AC*. //, therefore, we are given further that F(x) is continuous on E, then F(x) is ACG* on E. 236 CHAPTER VII. Functions of generalized bounded variation. Proof. It is enough to show that if at every point x of a set A the two extreme right-hand derivates F*(x) and F + (x) are finite, then A is expressible as the sum of an at mot enumerable infinity of sets on each of which the function F is A(\. Let A n denote, for each positive integer n, the set of the points xeA such that, for every <, (10.7) 0<t x^l/n implies \F(t)F(x)\^n-(t~x): and, for each integer i, let A' n denote the common part of A n and 00 OO -j- of the interval [i/n,(t+l)/n]. Plainly A=A n = A l n . n-^l //I / - oo Now, if I =[#i,# 2 ] is any interval whose end-points belong to A' n , we have, for every tel, the inequality 0^< x l ^.l/n, and so, on account of (10.7), \F(t)F(x l )\^n-(t j^Xn-f/l. This gives us O(-F;/)^2n-|Z|; and therefore for any finite sequence <!/} of such intervals, O(F;I/)^2n-2/Vyl- It follows that the function F j j is AC, on each of the sets J.J,, and this completes the proof. Theorem 10.5 shows, in particular, that every function which is continuous and everywhere derivable (even only unilaterally) is ACG*. Nevertheless as we saw in Chap. VI, p. 187, such a function need not be absolutely continuous. In view of Theorem 7.2, we may state also the following corollary of Theo- rems 10.1 and 10.5: A function F which fulfils at each point of a set E one at least of the inequalities (10.2) or (10.6), is derivable at almost all points of E. In part- icular therefore, the set of the points at which a function has (on one side at least) its derivative infinite, is of measure zero. These statements will be generalized in Chap. IX, 4. Theorems 10.1 and 10.5 contain sufficient conditions in order that a func- tion be VBG* or ACG*, but these conditions are clearly not necessary. Never- theless, by employing the notion of derivates relative to a function (cf. Chap. IV, p. 108), it is easy to establish conditions similar to those of the preceding theorems, the conditions being this time both sufficient and necessary. Thus, as shown by A. J. Ward [3]: In order that a finite function F be VBG* on a set E, it is necessary and suf- ficient that there exist a bounded increasing function U such that the extreme deriv- ates of F with respect to U are finite at each point of E except, perhaps, those of an enumerable set. 1 In order to establish the necessity cf the condition, let us suppose first that the function F is VB* on E. In view of Theorem 7.1 we may assume that the set E is bounded and closed. Let [a, b] be the smallest interval containing E, and, for each point x of the interval [a, 6], let V^x) and V t (x) denote the strong [ 10] Criteria for the classes of functions VBG*. ACG*, VBG and ACG. 237 variations of F (cf. 7) on the parts of E contained in the intervals [a, a;] and [x,b] respectively. Finally for each x of [a, 6], let V(x)==V l (x) V t (x)-}-x. The function V thus defined is increasing and finite on [a, 6], and can therefore be continued as a bounded increasing function on the whole straight line li^ We see at once that throughout the set E, except at most at the points a and 6, the derivates of the function F with respect to V are finite and indeed cannot ex- ceed in absolute value the number 1. Suppose now given any function F which is VBG* on E. The set E is then expressible as the sum of a sequence {E n } of sets on each of which the function F ib VB*. Consequently, by what has just been proved, there exists for each n a bounded increasing function V n with respect to which the function F possesses finite derivates at each point of the set E n except at most at the bounds of this set. Therefore, denoting by Mn the upper bound of |F/i(a;)| and writing U(x)=]Vn(x)/2 n Mn, we see at once that the function U thus defined is in- n creasing and bounded and that at each point of E, except perhaps those of an enumerable set, the function F possesses finite derivates with respect to U. 2 The condition is sufficient. Let F be a finite function having at each point of E, except perhaps at those of an enumerable subset, finite derivates with respect to a bounded increasing function U. For each positive integer n, let E n denote the set of the points x of E for which the inequality t x\-^.l/n implies \F(t) F(x)\-^.n*\U(t) U(x)\; and let each E n be expressed as the sum of a sequence {E l n }t=* ',2, .. of sets of diameter less than 1/n. We see easily (as in the proof of Theorem 10.1) that the function F is VB* on each set E r n , and since the sets E l n plainly cover all but an enumerable subset of E, it follows at once that the function F is VBG* on E. This completes the proof. If we analyze the first part of the above argument, we see that if the func- tion F is VBG* on E and moreover bounded on an interval containing the set E in its interior, there exists an increasing bounded function U with respect to which the function F has its derivates finite at each point of E. Moreover, if the function F is continuous on an interval containing E in its interior, the func- tion U may be defined in such a way as to be itself continuous (cf. the proof of Lemma 3.4, Chap. VIII). Finally, it can be shown that in order that a func- tion F be ACG* on an open interval I, it is necessary and sufficient that there exist an increasing and absolutely continuous function with respect to which the function F has its derivates finite at every point of I. (10.8) Theorem. If at every point x of a set E, except perhaps at the points of an enumerable subset, a function F fulfils any one of the inequalities (10.9) F+(x)<+ (10.10) then F is VBG on E. 238 CHAPTER VII. Functions of generalized bounded variation. Proof. We need only consider the case of the first of the inequalities (10.9) and that of the first of the inequalities (10.10). It is therefore sufficient to show that each of the sets A = E \F^ (x) < + oo] and B = B [F^(x) < -+- oo] is expressible as X X the sum of an enumerable infinity of sets on each of which F is of bounded variation. Consider first the set A. Given any positive integer n, let A n denote the set of all the points xe A such that, for every t, (10.11) O^tx^l/n implies F(t)F(x)*^n-(tx), and by A* n , for each integer f, the part of A n contained in the interval [i/n, (f+l)/n]. Let F n (x)=F(x)nx. For every pair x lt x% of points of A' n , where x l ^x 2J we have o > by (10.11), F(x 2 ) F(x 1 )^n-(x 2 x), i.e. The function F n (x) is thus monotone non-in- creasing on each set A l n , and it follows that A l n is expressible as the sum of a sequence of sets {An J }j=i&... on each which F n (x) is mono- tone and bounded. The function F(x)=F n (x)+nx is then plainly of bounded variation on each of the sets A l ^ J , and moreover we have 00 00 -1-00 00 a n-1 n-^i /=-oo;-=l Consider now the set B. From the definitions of approximate upper limit and approximate upper derivate (cf. 3, p. 220), it follows at once that to each point xeB we can make correspond positive integer n such that the set E -^j LJ_^ n has the I L x J point x as a point of dispersion. Therefore, denoting by B n the set of the points xeB such that the inequality O^fe^l/w implies both the inequalities (10.12) and (10.13) 00 f we have B = Bn- We denote further, for every integer t, by B n n~ == \ the part of B n contained in the interval [i/n, (t"+l)/n) and we write a& before, F n (x)F(x)nx. [ 10] Criteria for the classes of functions VBG,, ACG*, VBG and ACG. 239 The main part of the proof consists in showing that, for every i, the function F n (x) is monotone on B l n . For this purpose, let x v x 2 be any pair of points of a B l n , and let x l <x 2 . We plainly have 0<# 2 x^l/n, so that by writing x=x l and h=x 2 x l in (10.12), we obtain Similarly, from (10.13) with x=x 2 and fe=# 2 # 1? we derive The two inequalities thus obtained show that the interval ] contains a point such that and By adding these two inequalities term by term, we obtain F(x 2 )F(x l )<n-(x 2 x l ) J and so finally F n (x 2 )F n (x l )<Q. We have thus shown that the function F n (x) is monotone de- creasing on each B l n . It follows that B l n is expressible as the sum of a se- quence of sets {B'n J }j^\,2 f ... on each of which F n (x) is monotone and bounded, and on which the function F(x) = F n (x) + nx is ^therefore oo oo ~(-oo oo of bounded variation. Moreover, we have BB n ~ ^Ba J . n\ /!- 1 / oo j-.\ This completes the proof. On account of Theorem 4.2, it follows immediately from Theorem 10.8 that any measurable junction which satisfies one of the inequalities (10.9) or (10.10) at each point of a set E, is approximately derivable at almost all points of E. This proposition will be generalized and completed in Chap. IX ( 9 and 10). (10.14) Theorem. If two extreme approximate derivates on the same side are finite for a function F(x) at every point of a set E, except at most in an enumerable subset, then the set E is the sum of a sequence of sets on each of which F(x) is absolutely continuous. Consequently, if the function F(x) is further given to be contin- uous on E, then F(x) is ACG on E. Proof. It is clearly enough to show, for instance, that the set A=E[ oo < Ftp(x) ^. F^(x) <. +00] is the sum of an at most X enumerable infinity of sets on each of which F is AC. 240 CHAPTER VII. Functions of generalized bounded variation. Now we can make correspond, to each point X A, a positive integer n such that a? is a point of dispersion for the set E[\F()F(x)\^n-(t;x)] (cf. 3, p. 220). Hence, denoting by A n the set of the points X A such that, for every fe, the inequality ^ h ^ 2/n implies (10.15) we have A=A n ; and, denoting as before by A l n (for each integer i) n=-\ the part of A n contained in the interval [t/w, (t+l)/n], we obtain 00 -f-00 t n^\ / oo Consider now any two points x l and x 2 of A l n , where x l and let # 3 =2# 2 x t . We have, on the one hand, 0<# 3 x l 2-(x 3 x 2 )^.2/n, so that by writing x=x 1 and h = x 3 x l in (10.15), we obtain the inequality lEtlFU)-!^)!^^ and a fortiori (10.16) \E[\F(f)F(x l )\^n'(fx l )' J <p a < <# 3 ]|< (x 3 -x 2 )/2. On the other hand, we have 0^# 3 x 2 =x 2 x l ^.l/n j and so by (10.15) with x=x 2 and h=x 3 x 2 , we find (10.17) |E[ll^(c)-J f (* a )|>^^ The inequalities (10.16) and (10.17) show that there exists a point in [x 2 , o? 3 ] such that we have at the same time and this requires \F(x 2 )F(x l )\<4:n'\x 2 x l \. This last inequality is thus established for every pair of points x^x 2 of any one of the gets A* n , and it follows at once that F is AC on each of the sets A' n . This completes the proof. Theorem 10.8 shows in particular that a continuous function which is everywhere approximately derivable, even unilaterally, is necessarily ACG. In Chap. IX, 9, we shall give two further criteria for a function to be ACG* or ACG. CHAPTER VIII. Denjoy integrals. 1. Descriptive definition of the Denjoy integrals* We shall base the study of the Denjoy integrals on their descriptive definition. The essential ideas have already been sketched in Chap. VII, 1. We now complete them further as follows. A function of a real variable / will be termed S-intcgrable on an interval I=[a,b] if there exists a function F which is ACG on / and which has / for its approximate derivative almost every- where. The function F is then called indefinite ^-integral of / on /. Its increment F(I) = F(b) F(a) over the interval I is termed definite ^-integral of f over I and is denoted by or ff(x) dx. Similarly, a function / will be termed ^-integrable on an interval / = [a,&], if there exists a function F which is ACG, on I and which has / for its ordinary derivative almost everywhere. The function F is then called indefinite ^-integral of / on I; the difference FlI)=F(b)F(a) is termed definite ^-integral of / over / b and denoted by (^)ff(x)dx or by (&+) J f(x)dx. 7 ft For uniformity of notation, the Lebesgue integral will fre- quently be called -integral. The integrals 9 and \ are often given the names of Den joy integrals in the wide sense, and in the restricted sense, respectively. The first of these is also termed Denjoy-Khintchine integral (cf. Chap. VII, 1), and the second, Denjoy-Perron integral (for the latter, as we shall see below in 3, is equivalent to the Perron in- tegral considered in Chap. VI). 242 CHAPTER VIII. Denjoy integrals. It is immediate, by Theorem 6.2, Chap. VII, that when a func- tion is S- or SVintegrable on an interval, its definite Denjoy integrals are uniquely determined on this interval (its indefinite integrals being determined except for an additive constant). More generally, if two functions are equal almost everywhere and the one is integrable in the Denjoy sense (wide or restricted) on an interval I , then so is the other and the two functions have the same definite in- tegral over I Q . Another immediate consequence of the preceding definitions is the distributive property for Denjoy integrals. Thus, if two functions g and h are Q- or &+'integrable on an interval /, the same is true of any linear combination ag + bh of these functions, and we have It follows from Theorem 10.14, Chap. VII, that a continuous function which is approximately derivable at all points except, perhaps, at those of an enumerable set, is necessarily an indefinite ^-integral of its approximate derivative. Similarly, by Theorem 10.5. Chap. VII, a continuous function which is derivable (in the ordinary sense) at all but an enumerable set of points, is an indefinite & ^-integral of its derivative. The process of integration $ therefore includes that of Newton (cf. Chap. VI, 1). The fundamental relations between the Denjoy and Lebesgue processes are given in the following (1.1) Theorem. 1 A function f which is +-integrable on an interval I is necessarily also 9 -integrable on I and we have i / 2 A function f which is ^integrable on an interval I is necessarily +-integrable on I and we have ()+) fdx= fdx. / i 3 A junction which is >-integrablc and almost everywhere non- negative on an interval I is necessarily ~intcgrable on I. Proof. 1 and 2 follow at once from the definitions of tho Denjoy integrals and from the descriptive definition of the Lebes- gue integral (Chap. VII, 1). As regards 3, it is sufficient to recall the fact that, in view of Theorem 0.2, ('hap. VII, a function which IB ACG and whose approximate derivative is almost everywhere non-negative, IB necessarily monotone non-decreasing, and therefore it* derivative is Hummable. [ 1] Descriptive definition of the Denjoy integrals. 243 Part 3 of Theorem 1.1 shows that for functions of constant sign the Denjoy processes are equivalent to that of Lebesgue (cf. Theorem 6.5, Chap. VI, for the corresponding result concerning the Perron integral). Hence, we derive the following further extension of Lebesgue's theorem on term by term integration of monotone sequences of functions (Chap. I, Theorem 12.6). (1.2) Theorem. Given a non-decreasing sequence {f n } of functions which are ^-integrable on an interval I and whose ^-integrals over I constitute a sequence bounded above, the junction f(x) = lim f n (x) is n itself, necessarily, -integrable on I and we have () ff(x) dx = lim (S))ff n (x) dx. Exactly the same is true with &+ in place of 2) in the hypothesis and conclusion. Proof. This theorem reduces at once to the theorem of Le- besgue just referred to, for we need only consider in place of the functions / n , the functions //,, which are integrable in the Denjoy sense and non-negative, and which are therefore integrable in the Lebesgue sense on account of Theorem 1.1 (3). We shall show later on (Chap. IX, 11) that the extreme approximate derivates of any measurable function are themselves measurable functions. This includes the result that any function which is S-integrable is measurable. In the meantime we give an independent proof of this last assertion. (1.3) Theorem. A function which is 9-integrable is necessarily measurable and almost everywhere finite. Proof. Let / be S)-integrable on an interval I and let F be its indefinite integral. The function F is therefore ACG on /, so that I is the sum of a sequence {E n } of closed sets on each of which F is AC. By Lemma 4.1, Chap. VII, there exists for each n a func- tion F n of bounded variation on /, which coincides with F on E n . We therefore have almost everywhere on E n the relation f(x)=F ap (x]=F' n (x}j and since the derivative of a function of bounded variation is meas- urable and almost everywhere finite, it follows that / is!, meas- urable and almost everywhere finite on each E n and consequently on the whole interval I. 244 CHAPTER VIII. Denjoy integrals. Finally, let us mention as an immediate consequence of The- orem 9.1, Chap. VII, (1.4) Theorem. If a function f is 9-integrable on an interval Z , then every closed subset of J contains a portion Q such that the function f is summable on Q and such that the series of the definite ^-integrals of f over the intervals contiguous to Q is absolutely convergent. Similarly, if the function f is S> ^integrable on J , then every closed subset of I contains a portion Q such that the function f is sum- mable on Q and such that the series of the oscillations of the indefinite t-integrals of f over the intervals contiguous to Q is convergent. 2. Integration by parts* We have already observed (Chap. VI 7 p. 210) that a slight modification of the definition of Lebesgue-Stieltjes integral leads to an indefinite integral which is an additive function of an interval. As this modification will be useful to us in the present , we now formulate it explicitly. Given a finite function g integrable in the Lebesgue-Stieltjes sense with respect to a function of bounded variation F on an interval I = [o,b], we shall write b b (S)fg dF=fgAF-{g(a) \_F(a) -F(a-)]+g(b) [F(b+)-F(b)]}. a a b The number ($) j gdF will be called definite S-integral of g with a respect to F over I. As we see at once, this number (unlike the Lebesgue-Stieltjes integral) does not depend on the values taken by the function F outside the interval /, and for each point c of [a, b] we have (2.1) Theorem. Let g be a bounded function integrable with respect to a monotone non-decreasing function F on an interval [a, 6]. Then: b (i) (S)JgdFff[F(b)F(a)], where ^ is a number between a the bounds of the function g on [a, 6]; X (ii) writing 8(x) = (S) gdF for a^x^b, we have $'(#) = a = g(x)-F'(x) at almost all points of continuity of the function 0, and in fact at every point x where g is continuous and F derivable. [2] Integration by parts. 245 Proof. Clearly (i) follows at once from the obvious inequality b m'[F(b)F(a)]^(S)fgdF^M'[F(b)~F(a)] J where m and M are a the lower and upper bounds of g on [a,fc]. In order to establish (ii), consider a point x at which g is continuous and F is derivable. We may suppose, by subtracting a constant from g if necessary, that <7(# ) = 0. Denoting, for each interval J, by e(J) the upper bound of \g(x)\ on </, we have \S(J)\f\J\^8(J)-F(J)l\J\ and taking J to be an interval containing X Q and of length tending to zero, we find S'(x ) = Q = g(x ). This completes the proof. (2.2) Lemma. Let F be a function of bounded variation on an interval J [a,6], G a continuous function on / , and U the function defin- ed on I Q by the formula X (2.3) H(x)=F(x)G(x)(S)(G(t)dF(t) for a^x^b. a Then, if the function G is ACG [ACG*] on Z , so is the function H. Proof. We may clearly assume F to be monotone non-de- creasing. Denoting by M the upper bound of \F(x)\ on 7 , we shall begin by proving that for every interval /C^o we mu st have (2.4) \H(I)\^M 9 -\G(I)\+0(G- 9 I)-F(I) and O(ff; IK3Jf -O((?; I). In fact, by Theorem 2.1 (i), we have, for every subinterval J=[a, ft] of I, [G(p) -G(aft.F(p) + [F(p) -F(a)]-G(a) -(S)G(t) dF(t) where /z is a number between the bounds of G on J. Consequently, \H(J)\^Mf\G(J)\ + O(G- 9 J)-F(J) 9 and the first of the relations (2.4) follows by choosing J = I. On the other hand, we derive \H(J)\^3Mo-O(G; I) for every interval JCI, and hence the second relation (2.4). Hence, since the function G is continuous, the function H is continuous also. Further, if {I k } is any finite sequence of non- overlapping intervals and if o> denotes the largest of the numbers O(<?; / A ), we obtain from the relations (2.4) and 246 CHAPTER VIII. Denjoy integrals. The first of these inequalities implies that if the function G is AC on a set E, so is the function H, and consequently, that if the func- tion G is ACG on the whole interval I , then the function H is also ACG on J . Similarly, the second of the above inequalities shows that if G is ACG,,, on / then so is F, and this completes the proof. We can now complete Theorem 14.8, Chap. Ill, which concerned integration by parts for the Lebesgue integral, by establishing a similar theorem for the Denjoy integrals: (2.5) Theorem. If F(x) is a junction of bounded variation and g(x) a junction 9- or &+-integrable on an interval 1 = [a, b], then the function F(x) g(x) is integrable on I Q in the same sense, and moreover denoting by G the indefinite integral of g, we have h b (&)fF(x)g(x)dx=G(b)F(b)-G(a)F(a)-(S) fG(x)dF(x). a a Proof. We shall establish the theorem for the ^-integral. The proof for the ^-integral is quite similar. By Lemma 2.2, the function H defined by the formula (2.3) is ACG on / . Moreover, by Theorem 2.1 (ii), if we form the ap- proximate derivative of both sides of (2.3), we obtain almost every- where the relation H f ^(x)^=F(x)G' aip (x)==F(x)g(x). It follows that the function F(x)g(x) is 9-integrable on the interval / and that b ()JF(x) g(x) dx^=H(b)~- H(a). This" last relation is equivalent to a the one to be proved. The idea of the above proof, which is directly based on the descriptive defi- nition of the Donjoy integrals, is due to Zygmund. For another proof, depending on the constructive definition of these integrals, cf. for instance E. W. Hobson [I, p. 711]. For an interesting generalization of the theorem on integration by parts to the c/S-integral (cf. Chap. VI, 8) vide A. J. Ward [3]. From Theorem 2.5, there follows easily the second mean value theorem for the Denjoy integral, which may be regarded as a generalization of Theorem 14.10, Chap. III. (2.6) Theorem. Given a non-decreasing function F on an interval I [a,b] and a function g which is -integrable on I , there must exist a point in / such that (S>)fg(x)F(x)dx=F(a)'(S))fg(x) [ 3] Theorem of Hake-Alexandroft-Looman. 247 Proof. Writing G(x) = ($) f g(t)dt, we have by Theorems 2.5 a and 2.1 (i) the relation b b (9) fg(x)F(x) dx = G(b)F(b) - (S)fO(x) dF(x) = where // is a number between the bounds of G(x) on /. It follows that there exists a point in I a such that /*={?(), and the rela- tion just obtained becomes b (S>)fg(x)F(x)dx=F(a)-G(S)+F(b).[G(b)-G()], a which, by definition of G(x), reduces to the required formula. 3. Theorem of Hake-Alexandroff-Looman. The relations between the Den joy integrals and those of Lebesgue and of Newton having already been obtained in 1, we now proceed to establish an important result of Hake, Alexandroff and Looman, which asserts the equivalence of integration in the restricted Den joy sense with Perron integration. At the same time we shall show that in the definition of Perron integral (Chap. VI, 6) we need only take account of the continuous major and minor functions. In order to make this assertion quite precise, let us agree to say that a function / is ^-integrable on an interval / if 1 the function has continuous major and minor func- tions on I and 2 denoting by U any continuous major function and by V any continuous minor function of / on 7 , the lower bound of the numbers U(I ) is equal to the upper bound of the numbers F(/ ). The function / is then plainly ^-integrable on 7 , the definite ^-integral of / on I being equal to this common bound. We have to prove the converse, i. e. that every function which is o^-inte- grable is also ^-integrable. (3.1) Lemma. If a function f is <P Q -integrable on each interval interior to an interval [a, 6] and if the definite ^-integral over the interval [a + e,ft rj] tends to a finite limit as e >0 + and ?^->0+, then the function f is ^-integrable on the whole interval [a, ft]. 248 CHAPTER VIII, Denjoy integrals. Proof. It is clearly sufficient (by halving the given interval) to consider the case of a function / which is <^ -integrable on each X interval of the form [a,* e] where 0<e<& a. Let P(x)=(&)lfdx for a^x<b and p = P(b ). We choose any positive number cr. Writing for symmetry a =a, we consider any increasing sequence of points {^ k } k=0 , which con- verges to ft. The function / being <^ -integrable on each interval [a>k> flJH-i]> we easily define, on the half open interval [a, ft), a con- tinuous function F such that F is a major function of / on each of the intervals [a/,, a A +,] and that [F(x) F(a k )] [P(x) P(a k )]<a/2* for ak^x^.aki and %=0,1, .... By the second of these conditions the oscillation of the function F on the interval [a*, 6) tends to as fc->oo, and therefore F has a finite limit F(b-) at the point b. Writing F(x)=F(a) + (x a) 13 for x<a, and F(x)=F(b) for o?>ft, we extend the definition of F to make this function continuous on the whole straight line R^ and the following conditions are then satisfied: (3.2) oo^F(x)^f(x) for a^x<b and (3.3) F(b) Now let c be an interior point of [a, b] such that the oscillation of F on [c,ft] is less than a. For each point x of [c,ft], let 0(x) denote the oscillation of F on [x, b]. The function 0(x) is continuous and non-increasing on the interval [c,ft], and we extend its definition on to the whole straight line R^ by making 0(x) = 0(c) for#<c and 0(x)=0(b) = for x>b. We now write Q(x)=F(x)0(x) and U (x) = G (x) + a (x - &) I/3 / (b - a) 1 3 . Since the function 0-(x-- 6) 13 /(6 a) 1 '* 0(x) is non-decreasing, it follows at once from (3.2) that oo=U(x) ^f (x) for a^x<b. Moreover, since Q(b) 0(x) is non-negative for each point x of the interval [c,fe], and for x^bj we find 6(6)^0, and therefore U(b)+oo. Hence U is a con- tinuous major function of / on the interval [a, ft] and fulfils, by (3.3), the inequality U(b) U(a)^F(b) F(a)+ 2<7<p+4<r. Similarly we define a function V which is a continuous minor function of / on [a,6] and fulfils the condition F(ft) V(a)^p 4a. It follows that the function / is ^-integrable on [a, ft]. This completes the proof. [ 3] Theorem of Hake-Alexandroff -Looman. 249 (3.4) Lemma. Let Q be a closed .cmd bounded set, a, b its bounds, {Ik[ak, bk]} the sequence of intervals contiguous to Q, and f a function which is summable on Q and t? -integrable on each interval contiguous to Q. Then, if the series of the oscillations of the indefinite ^-integrals of the function f on the intervals Ik converges, the function f is & Q -integrable on the whole interval [a,fc] and we have (3.5) Proof. Let e be a positive number and let K be a positive integer such that oo (3.6) Z0 k <e, k--K\ \ where Ok denotes the oscillation of the indefinite o/Mntegral of / on the interval I k . Denote by f l the function which agrees with / on the set Q and on the intervals Ik for k^K, and which is elsewhere. By Theorem 3.2, Chap. VI, and by the hypotheses of the lemma, the function f 1 has a continuous major function U 1 and a continuous minor function V l such that (3.7) We shall now define a continuous major function for / / x . Let Fk be, for each A:, a continuous major function of / on the interval I k auchthat F k (a*) = and O(F k ; I k )^20 k , and let A k (x) and Bk(x) denote, for any point #e/*, the oscillations of the func- tion Fk on the intervals [a*, x] and [#,&*] respectively. We write G(x) = Fk(x) + A k (x)[Bk(x)B fi (ak)\ when xel? and k>K, and Q(x) = Q elsewhere. Finally, for each a?, we write where the summation (Jc) ii> extended over the indices k for which &*<#. Since, for every fc, we have G(a k +) = G(ak) = Q and O((r;I^)<3-O(jPA; /ArXG-O/r, the function E7 2 is continuous on the straight line K^ and since the function tf vanishes identically on each interval Ik for fc<#, we have by (3.6), (3.8) 250 CHAPTER VIII. Denjoy integrals. Now, for each fc, we have (?(#)^0 and G(b k ) (?(#)^0 for every point xtl*. Therefore the increment of the function U 2 is non-negative on each interval containing points of the set Qj and consequently L^M^^/M/!^) at eac h point x of this set. Again, since the function G vanishes on each interval I k for k^K, we have C/ 2 (.r) = = /(#) -/^tf), whenever xell for k^K. Finally, since the function A h (x) B k (x) is non-decreasing on each I*, we- see that tt^U 2 (x)^Fti(tt)l^f(z)==f(%)-fi(z) at each point #eZJ for k>K. Thus U 2 is a continuous major function of ffi on [a, 6], Similarly, we determine a continuous minor function F 2 of / / 1; subject to the condition F 2 (&) F 2 (a)>6e which corresponds to (3.8). Therefore, writing U^U^U^ and V^V^ + V^ we obtain a continuous major function U and a continuous minor function V for / on [a, ft], and if we denote by p the right-hand side of (3.5), we obtain from (3.6) and (3.7), U(b) U(a) 8e^p<F(fr) V(a)+Se. The function / is thus #J>-integrable on the interval [a, b] and its definite ^-integral over this interval is given by the formula (3.5). (3.9) Theorem. A function f which is $+-integrable on an interval Z is necessarily & Q -integrable on / , and we have Proof. Let F be an indefinite ^-integral of / on I . We call an interval IC_I regular, if the function / is ^-integrable on I and if the function F is on I an indefinite ^-integral of /. Further, we call a point #e/ regular, if each sufficiently small interval /C^o containing x is regular. Let P be the set of the non-regular points of I . We see at once that the set P is closed and that every subinterval of I which contains no points of this set is regular. We have to prove that the set P is empty. Suppose, if possible, that P4=0. By Lemma 3.1 we see easily that every interval contiguous to P is regular and that the set P theretore has no isolated points. On the other hand, by Theorem 9.1, Chap. VII, the set P contains a portion P on which the function F is AC,. Let J be the smallest interval containing P . Since the set P has no isolated points, the same is true of any portion of P, and therefore P-Jo={=0. It follows that in order to obtain a con- tradiction, which will justify our assertion, we need only prove that the interval J is regular. [3] Theorem of Hake-Alexandroff-Looman. 251 To show this, let J be any subinterval of J and let Q be the set consisting of the points of the set P-J and of the end-points of J. We denote by {/} the sequence of the intervals contiguous to Q and by G the function which coincides with F on Q and is linear on the intervals I n . Plainly the function G is absolutely continuous on J. Therefore, since G'(x) = F'(x) = f(x) at almost all points x of Q, and since G(I n ) F(I n ) for each n, we obtain (3.10) Now the function / is summable on Q and ^-integrable on each interval / and moreover, F is an indefinite c/Mntegral of / on each of these intervals. The series of the oscillations of F on the intervals I n being convergent, it follows, by Lemma 3.4, that the function / is ^-integrable on J and that, on account of (3.10), F(J) = (&) ffdx. Therefore, since J is any subinterval of J Q , the j interval J is regular and this completes the proof. (3.11) Theorem. A junction which is ^-integrable on an interval I is necessarily &+-integrable on I . Proof. Let / be a function ^-integrable on an interval / and let P be its indefinite ^-integral. We shall show that the func- tion P is an indefinite >* -integral of /. Since P'(x) = f(x) almost everywhere (cf. Theorem 6.1, Chap. VI), it is enough to show that the function P is ACG* on / , i. e. that any closed set QC_I con- tains a portion on which the function P is AC^. Let H be any major function of /, Since E[(x)> oo at each point x of I , the function H is by Theorem 10.1, Chap. VII, VBG, on I , and hence Z is expressible as the sum of a sequence of closed sets (cf. Theorem 7.1, Chap. VII) on each of which the function U is VB*. It follows, by Baire's Theorem (Theorem 9.2, Chap. II) that the set Q contains a portion Q Q on which the function H is VB^. Since the difference P H is a monotone function, the function P is actually VB^ on Q Q . We shall show that P is further AC* on $ . For this purpose, we denote by J [a, 6] the smallest interval containing Q Q . Let e be any positive number and U a major func- tion of / on / such that (3.12) 252 CHAPTER VIII. Denjoy integrate. Let P l and U^ denote the functions which coincide on (? with the functions P and U respectively, and which are linear on the intervals contiguous to Q and constant on the half -lines ( oo, a] and [&,-}-oo). The function P l is clearly of bounded variation. On the other hand, we see easily that U l (x)>oo at every point, and that U 1 (x)^P'}(x) at almost all points, of the interval J . Therefore, writing f^^P'^x) wherever the second of the above inequalities holds, and f l (x) = -~oo elsewhere, we see at once that the function / t (j?) is sunimable on J and has V l for a major function. It follows that Ui(I)^ f fi(x)dx i for each interval I(2J QJ and therefore that the function of singularities (cf. Chap. IV, p. .120) of U l is monotone non-decreasing on J . Let T l be the function of singularities of P. Since the function P l U^ is monotone non-increasing on J and since, by (3.12), we have Q^P l (J )-U l (J )=P(J )-U(J )'^ e, it follows that l\(I)^-e for each interval /C^o? an( i being any positive number, this requires I\(I)>0 for every interval /C^o- Similarly, by considering minor functions of / in place of major functions, we find 2\(/)^0, and therefore, finally, f l\(I) ^0, for each interval /C^o- Th e func- tion Pj is thus absolutely continuous on J . This requires the func- tion P to be AC on the set Q Q as well as VB + , and therefore AC^ on this set on account of Theorem 8.8, Chap. VII. Thus every clos- ed set QCIo contains a portion Q {} on whiclj the function P is ^, and this completes the proof. The first of the theorems proved in this , which together establish the equivalence of the processes of t *V, >^ - and ^-integration, was derived in 1921 by II. Hake [1] from the constructive definition of the integral $* (vide below, 6). The second theorem was obtained some years later by P. Alex- androff [1; 2] and II. Loo man [4] independently. For an interesting extension of these results to Perron-Stieltjes integral, vide A. J. Ward [3]. It should, perhaps, be added that in their original definitions 0. Per- ron [1] and 0. Bauer [1] employed only continuous major and minor functions. The equivalence of the original Perron -Bauer definition with that of Chap. VI, 6, has therefore been established here as a consequence of Theorems 3.9 et 3.11. Let us remark further that in the definition of Perron integral, ordinary major and minor functions may be replaced by generalized continuous major and minor functions defined as follows. A function U is a generalized continuous major function of a function / on an interval I if 1 U is continuous and VBG on I, 2 the set of the values assumed by U at the points at which U'(x) oo, is of measure zero, and 3 U(x)^f(x) at almost all points x. The definition of generalized continuous minor functions is obtained by symmetry. [ 3] Theorem of Hake-Alexandroff-Looman. 253 We shall conclude this with the following result, due to Marcinkiewicz: (3.13) Theorem. A measurable function f which has on I at least one continuous major function and at least one continuous minor function, is necessarily &-integrable on I . Proof. Let U and V be respectively a continuous major func- tion and a continuous minor function of / on I . We shall call a point xeI regular if the function / is <^-integrable on each suf- ficiently small interval /QZo which contains x. Let Q be the set of the points x of / which are not regular. The set Q is plainly closed and we see at once that the function / is <#Mntegrable on each sub- interval of J which contains no points of Q. Thus it has to be proved that $ 0. Suppose, if possible, that Q=^Q. For every interval I on which the function / is <Mntegrable, we have U(I)^(c?) ff(x)dx^V(I). i Therefore, if [a,b] is an interval contiguous to Q, the definite ^-integral of / on the interval [a + e, b rj~] interior to [a,&] tends to a finite limit as ->0 and 77^0. By Lemma 3.1, the function / is thus c^-integrable on each interval contiguous to Q. It follows, in particular, that Q can have no isolated points. Now let Q Q be a portion of Q on which the functions U and V are both VB*. Such a portion exists by Theorem 9.1, Chap. VII, since the functions J7and Fare VBG* on I on account of Theorem 10.1, Chap. VII. Let J be the smallest interval containing Q . Since U.(x)^f(x)^ V(x) everywhere on Z , the function / is summable on Q together with the two derivatives U(x) and V(x). On the other hand, denoting by {!} the sequence of the intervals contiguous to Q and by O n the oscillation on I,, of the indefinite ^-integral of /, we shall have 0,,<O( [/;!) + O(F; /) for every n, and so n <+oo. It follows by Lemma 3.4, that the function / is <f-inte- 71 grable on the whole interval J . But this is clearly impossible, for since the set Q has no isolated points, the interval J contains in its interior some points of Q. We thus arrive at a contradiction which completes the proof. Just as in the definition of the Perron integral, we may replace, in The- orem 3.13, ordinary major and minor functions by generalized continuous ones (cf. above, p. 252). Nevertheless, the conditions of Theorem 3.13 differ from those of the definition of Chap. VI, 6, in that continuity is essential. In fact, if we write /(x) = for x^O and f(x) = 1/x* for z>0, U(x)=0 identically in J^ and V(x) = for x^O and V(x) l/x for x>0, we see at once that U and V are 254 CHAPTER VIII. Denjoy integrals. * 4. General notion of Integral. We shall deal in this with some notions of a more abstract kind which we shall employ, in the next , as a basis for the constructive definition of the Denjoy integrals. Let be a functional operation by which there corresponds to each interval I = [a,b] a class of functions defined on I, and to each function / of this class a finite real number. This class of func- tions will be called domain of the operation t on the interval I, and the number associated with / will be denoted by (/;!). An operation will be termed an integral, if the following three conditions are fulfilled: (i) If a function / belongs to the domain of the operation on an interval jf , the function belongs also to the domain of on any interval /C^o> an( i ^(/jJ) * s a continuous additive function of the interval /C^o- (ii) If a function / belongs to the domain of the operation on two abutting intervals jf x and I 2 , ^^ e function belongs also to the domain of on the interval Ii+I 2 . (iii) A function / which vanishes identically on an interval I belongs to the domain of on Z, and we have (/;!) 0. If C is an integral, any function / which belongs to the domain of on an interval / will be termed -integrable on / and the number (/;I ) will be called definite ^integral of / on I . The function of an interval IC^o? (/> ^)> which is additive and continuous on account of (i), will then be called indefinite fa-integral of / on / and its oscillation on / (i. e. the upper bound of the numbers |(/; I)| f where / denotes any subinterval of I ) will be denoted by O(; /; I ). Two integrals i and 2 will be termed compatible, if ^(/; I) = &z(fil) for every interval / and for every function / which is both C r and f 2 -integrable on /. We shall say that the integral 2 includes the integral 6^, if the two integrals are compatible and if every function which is ^integrable is also 2 -integrable. When this is so we shall write Given an integral and a function g which vanishes out- side a bounded set E, it is evident that if g is T-integrable on an interval j[ which contains E in its interior, then g is so also ou aay interval I which contains E, and we have G(g; I)= [4] General notion of integral. 255 This fact justifies the following definition: we shall say that a function / is -integrable on a bounded set E, if the function g which coincides- with / on E and is elsewhere, is -integrable on each interval I^)E. The number (gr;I) is then independent of the choice of the interval /D^5 we shall call this number definite ^-integral of the function / on the set E and we shall denote it by (/; E). Of the known processes of integration, all those which give rise to a con- tinuous indefinite integral (for instance those of Lebesgue, Newton, Den joy, etc.) are easily seen to be integrals according to the above definition. If, however^ we wished to include also discontinuous integrals (e.g. that of W. H. Young cf. Chap. VII, p. 215) we should have to modify some details of the definition! Given a function / on an interval / and given an integral C, we shall say that a point ae! Q is a ^-singular point of / in I Q if there exist arbitrarily small intervals /C^o containing a on each of which the function / is not <T-integrable. Denoting by 8 the set of these points, we see at once that the set 8 is closed and that the function / is c-integrable on every subinterval of / which contains no points of 8. Wi'th each integral we now associate three "generalized" integrals c , H and H *. defined as follows. Given any interval / , the domain of the operation C c on 7 is the class of all the functions / which fulfil the following two conditions: (c 1 ) the set of the ^-singular points of / in I is finite (or empty); (c 2 ) there exists a continuous additive function of an interval F on IQ such that JF (/)=(/; I) whenever I is a subinterval of I which contains no -singular point of /. Since such a function F (if existent) is uniquely determined by the conditions (c 1 ) and (c 2 ), we can write ( '(/; / )= The domain of the operation e H on / is defined as the class of the functions / which fulfil the following conditions: (h 1 ) if 8 denotes the set of all c- singular points of / in I , the function / is ^-integrable on the set 8 and on each of the intervals I k contiguous to the set consisting of the points of 8 and of the end- points of I ; (h 2 ) Z |(/; Ik)\< + and, in the case in which the sequence {I*} k is infinite, lira O(; /; !*) = 0. 256 CHAPTER VIII. Denjoy integrals. For any such function /, we write by definition: H (/; / ) =S *(/;!*) + (/; 8). Finally, we obtain the definition of the operation H * by re- placing in the definition of the operation <T H the condition (h 2 ) by the more restrictive condition: +00. r* H H We verify at once that the operations , and * all fulfil the conditions (i), (ii) and (iii), p. 254. These operations are there- fore integrals according to the definition, p. 254, and we evidently have C C and 2TC^ H *C^ H - F r brevity, we shall write (H and CH * in place of ( r ) H and (f ( ') !i * respectively. The integral c and the integrals C H and H * may be regarded respectively as the Cauchy and the Harnack generalizations of the integral . They correspond, in fact, to the classical processes employed by Cauchy and Harnack to extend integration from bounded to unbounded functions of certain classes. The original process of Harnack actually corresponds to the operation <T H * rather than to the operation H . Cf. A. Harnack [1], E. W. Hobson [I, Chap. VIII] and A. Rosen- thai [I, p. 1053]. If we were to add to the conditions (h 1 ) and (h 2 ) which characterize the generalized integral H , the condition that limO((T; /;/*)/(>(a&, 1^) for almost k all x $, we should arrive at a generalized integral C n intermediate between C H and C H *. By applying the process C 11 ' in the constructive definitions of Den- joy integrals of the next , we should then obtain an integral "?', intermediate between $ and ^. Its descriptive definition is very simple: a junction / is l ?'- integrable if it is $-integrable and if its indefinite ^-integral is almost everywhere derivable (in the ordinary sense). This integral has been discussed by A. Khin- tchine [1]; cf. also J. C. Burkill [1]. * 5. Constructive definition of the Denjoy integrals. With the notation of the preceding , we see at once that for each integral C^> w ^ have also C C; similarly the relation implies C*- I* * 8 n t quite so obvious that the relations and ^C* imply respectively H C and ' H *C ) *- This last as- sertion is a consequence of the following theorem which is anal- ogous to Lemma 3.4. [5] Constructive definition of the Denjoy integrals. 257 (5.1) Theorem. Let Q be a bounded closed set with the bounds a and 6, and let {I k } be the sequence of intervals contiguous to Q; and suppose that f is a function 9-integrable on the set Q as well as on each of the intervals I*, and that (in the case in which the sequence {Ik} is infinite} and * Then the function f is &-integrable on the whole interval ft] and we have (5.2) ( i Q k i k If we suppose, further, that the function f is &+-integrable on Q as well as on each of the intervals I k and that ^0(^5 /; //,)< +00, then the junction f is &+-integrable on I. k Proof. We shall prove the theorem for the ^-integral. The case of the 9, -integral is similar. Let I(x) denote the interval [a,x] where we suppose xe[a,b], and let (5.3) F(x)=S(S>) [f(*)dt. * V/w We shall show that the function F, thus defined, is ACG on the interval I. For this purpose, it will suffice to show that F is AC on the set Q, the function being evidently continuous on / and ACG on each of the intervals //,. Let g(x) be the function equal to for xeQ and to ry-r () / / (t)dt i z *i / for re el* where fc=l, 2, .... The function g is summable on / and JC if O(x) g(t) dt, the function F clearly coincides with G on Q; a F is thus AC on Q and therefore ACG on I. This being so, we have F r ^(x)==O'(x) = g(x)=^ at almost all points x of Q, while it follows at once from (5.3) that F^(x)=f(x) at almost all points x of I Q. Hence, F being ACG on /, it follows that the function equal to / on I Q and to on Q has F for an indefinite ^-integral. On the other hand, the function equal to / on Q and to elsewhere is, by hypothesis, 9-integrable on I. It follows that the function / itself is sMntegrable on I, and that (9)ff(x) dx=F(b) F(a) + (&) ffdx, which, on account of (5.3), / Q is equivalent to (5.2). This completes the proof. 258 CHAPTER VIII. Denjoy integrals. We now pass on to the constructive definition of the Denjoy integrals. We begin by introducing the following notation. Let {-} be a sequence of integrals, in general transfinite, such that ^C v whenever !<*?. We then denote by %* the operation whose domain on each interval I is the sum of the domains of the operations c 1 for f<a, and which is defined for every function / of its domain by the relation e(/; I)=~(f; I), where f is the least of the indices f < a such that / is ^-integrable on I. It then follows, of course, that (/; /)*(/; I) for every f > , since by hypothesis & then includes e-. This being so, let {*} and {;} be two transfinite sequences defined, by an induction starting with the Lebesgue integral f, as follows: * /or ?< < Denoting by Q the smallest ordinal number of the third class (cf. for instance, W. Sierpiriski [I, p. 235]) we shall show that 0= 2V = S Q and S. = Sfi - < l-'fi We shall restrict ourselves to the case of the ^-integral (that of the ^-integral being quite similar). Since C i w find at once by induction (cf. above, p. 256) that for every , *&, and so. obviously, 2VC I n order to |- Q change this last relation into one of identity, it is enough to show that every function / which is -integrable on an interval lQ=[a,b]j is TMntegrable on 7 for some index <?. Let 8^ denote the set of the ^-singular points of / in 7 . The sequence {S*}, as a descending sequence of closed sets, is sta- tionary, i.e. there exists an index v<& such that ST=S V+ *' (For if not, there would exist for every <Q a point x8*S* +l , and therefore also an interval 1$ with rational end- points, containing the point x* of S* but without points in common with the closed set S^\ nor therefore, with any of the sets $ l+2 , S*" 4 " 3 , ... . We should thus obtain a transfinite sequence of type Q of distinct intervals with rational end-points, and this is impossible.) We shall prove that $ y =0. [f 5] Constructive definition of the Den joy integrals. 259 Suppose, if possible, that S'^Q. We see at once that the func- tion / is >C"-integrable on each interval /C^o which contains no points of 8 V . It follows that the function / is (O c -inte,gnble, and a fortiori >C v+1 -integrable, on each interval contiguous to S". Since S v =8 v+l , it follows, in particular, that the set S v contains no iso- lated points. The function / being, by hypothesis, S>-integrable on I , the set S v (cf. Theorem 1.4) must contain a portion Q such that the function / is summable on Q and such that the scries of the definite ^-integrals of/ over the intervals contiguous to Q converges absolu- tely. Since C(^O C 'C^> it follows at once that the function / is ( v ) (Ji -integrable, i.e. T^-integrable, on some interval J contain- ing Q. But this is clearly impossible, since, in view of the fact that the set 8 V has no isolated points, the interval J certainly contains points ,of the set S v =8 v+l in its interior. We thus have 8 V =Q, which establishes the r"-integrabittty of / on 7 and completes the proof. Various definitions, constructive and descriptive, of Den joy integrals will be found in the papers mentioned in Chap. VI, p. 207, and Chap. VII, pp. 214-215, as well as in the following treatises and memoirs: N. Lusin [I; 4], T. II. Hil- debrandt [1], P. Nalli [I], E. Kamke [I], A. Kolmogoroff [2], H. Lebesgue [7; II, Chap. X], A. Rosenthal [1] and P. Romanowski [I]. For further extensions to functions of two or more variables, see alao H. Looman [1] and M. Krzyzanski [1]. CHAPTEE IX. Derivates of functions of one or two real variables. 1. Some elementary theorems. The first part of this chapter ( 110) is devoted to studying the various relations between the derivates of a function of a real variable. With the help of the no- tion of extreme differentials introduced by Haslam-Jones, certain of these relations will subsequently be extended, in the second part of the chapter ( 11 14), to functions of two variables. Accordingly, the term "function" will be restricted in the first part of this chapter to mean function of one real variable. Before proceeding to the theorems directly connected with the Lebesgue theory, we shall establish in this some elementary results. We first observe that a linear set E contains at most a finite num- ber, or an enumerable infinity, of points which are isolated on one side at least. To fix the ideas, let A be the set of the points of E which are isolated points of E on the right. For each integer n, let A n denote the set of the points x of A such that the interval [x,x+lln] contains no point of E other than x. Then it is plain that, for each integer fc, the in- terval [fc/n, (k+ I) /n] can have at most one point in common with A n . Hence each set A n is at most enumerable, and the same is true of the set A A n . n We say that a finite function F assumes at a point x a strict maximum if there exists an open interval / containing # such that F(x)<F(x ) for every point X I other than # . By symmetry we define a strict minimum. [1] Some elementary theorems. 261 (1.1) Theorem. Given a finite function of a real variable F, each of the following sets is at most enumerable: (i) the set of the points at which the function F assumes a strict maximum or minimum; (ii) the set of the points x at which lim sup JP 1 ()>lim sup F(t) or liminf F(t)<\im inf F(t); t-+x /->*+ t-+x /->*+ (iii) the set of the points x at which F+(x)<F~(x) or F~~(x)<F+(x). Proof, re (i). Consider the set A of the points at which, for in- stance, the function F assumes a strict maximum, and let A n denote, for each positive integer n, the set of the points x such that F(t)<F(x) holds for each point t^x of the interval (x 1/n, x+l/n). We see at once that each set A n is isolated, and therefore at most enumerable. Since A=A nj it follows that the set A is at most enumerable. re (ii). Let us consider, for definiteness, the set B of the points x at which limsupjF(J)>lim sup F(t). We denote, for each pair of in- t-*x t+x+ tegers p and <?, by R p>q the set of the points x such that lim sup F(t) > pjq > lim sup F(t). t-+x f-Jc-H Clearly each point of a set B p , q is, for that set, an isolated point on the right. Each of the sets B p , q is thus at most enumerable, and, since B~ B pt(n the same is true of the whole set B. P,q _ re (iii). Consider the set C of the points x at which F*(x)<J?~(x). and denote, for each pair of integers q>0 and p, by C p , q the set of the points x at which F + (x)<plq<I r ~(x). Write F p , Q (x)=F(x)px/q. We find F^ t g(x)<Q<F^ q (x) at each point xcC piqj and this shows that the function F p , q assumes a strict maximum at each point of C p , q . By the result just established, each set C p , q is at most enumerable, and con- sequently, the same is true of the whole set C. It is sometimes convenient (vide, below, 5) to appeal to a slightly more general form of the last part of Theorem 1.1, which concerns relative derivates (cf. Chap. IV, p. 108) and which reads thus: (1.2) Theorem. If U and F are two finite functions of a real variable, the set of the points t at which the derivative U'(t)>Q (finite or infinite) exists and at which F~u(t)<Fu(t), is at most enumerable. 262 CHAPTER IX. Derivates of functions of one or two real variables. This is proved in the same way as the corresponding part of Theoreml.l. In fact, if we denote, for every pair oHntegers g>0 and p, by CM the set of the points x at which F~u(x)<plq<Fu(x), we see at once that the function F(x)-(plq)-U(x) assumes at each point of C pq a strict maximum. Therefore each set C M is at most enumerable. For Theorem 1.1 and its various generalizations, vide: A. Den joy [1, p. 147], B Levi | !]. A. Rosenthal [1], A. Schonflies [I, p. 158], W. Sierpinski [1; 2] and Ci ( Y on up [ 1]. As regards the enumerability of the set of the points at which the function assumes a strict maximum or minimum, it is easily seen that this reMih remains valid for functions in any separable metrical space (cf. F. Ilaus- dorff 'J, p. 363]). Mention should be made also of the elegant generalizations of Theorem 1.1, obtained successively by H.Blumberg [1], M. Schmeiser [1] and V. Jarnik [3]. 2. Contingent of a set. We have mentioned earlier (in Chapter IV, p. 133), that certain theorems on derivates of functions may he stated as propositions concerning metrical properties of sets in Euclidean spaces. In connection with these results, we shall state in this some definitions which begin with some well-known notions of Analytical Geometry. Hy the direction of a half -line I in a space R m (where m^2) we shall mean the system of the m direction cosines of I. The half-line itfeuing from a point a and having the direction will l>e denoted by aft. The half-line issuing from a point a and containing a point b^=a will be denoted by ab. If we interpret the system of the m direction cosines of a half- line as a point in R m (situated on the surface of a unit sphere), we may regard the set of all directions in a Euclidean space as a complete, separable, metrical space (cf. Chap. II, 2). It is then clear what is to be understood by the terms: convergence and limit of a se- quence of directions, everywhere dense set of directions, etc. We shall say further that a sequence of half-lines {/} issuing from the same point a converges to a half -line I issuing from a, if the sequence of the directions of the half -lines l n converges to the direction of I. Given a set K in a space It m , a half-line I issuing from a point aeE will be called an intermediate half -tangent of K at a, if there exists a sequence \a n ] of points of K distinct from a, converging to a and such -> that the sequence of half -lines {aa n } converges to J. The set of all inter- mediate half-tangents of a set E at a point a is termed, following [2] Contingent of a set. 263 G. Bouligand [I], the contingent of E at a and denoted by contg^a (by the contingent of E at an isolated point of E, we shall understand the empty set). A straight line passing through a which is formed of two intermediate half-tangents of E at a is called intermediate tangent of E at a. Similarly a hyperplane h passing through the point a is called intermediate tangent hyperplane of E at a, if each half -line issuing from a and situated in h is an intermediate half-tangent of E at a. In R 2 the notions of intermediate tangent hyperplane and inter- mediate tangent are plainly equivalent. Given in the space R m a hyperplane A, a } x l + ax 2 i-...-\-a m x m =b< (cf. Chapter III, 2) the two half-spaces (half-planes if w = 2) a 1 # 1 + a 2 # 2 + ...+a m x m ^b and a 1 3? 1 + a 2 je 2 +...+mffiii<&, into which ft divides JR mj will be termed *ufes of the hyperplane fc. In the case in which h is an intermediate tangent hyperplane of a set E at a point a and in which, further, the contingent contg#a is wholly situated on one side of A, the side opposite to the latter is called empty side of h and the hyperplane h is termed extreme tangent hyperplane of E at a. The two sides of h may, of course, both be empty at the same time, and this occurs if the contingent of E at a coincides with the set of all half -lines issuing from a which lie in the hyperplane h itself. The hyperplane h is then termed unique tangent hyperplane, or simply, tangent hyperplane, of E at a. For simplicity of wording, we shall restrict ourselves in the sequel to the case of sets situated either in the plane JK 2 or in the space R 3 . Needless to say, the extension to any space R m presents no essential difficulty (an elegant statement, which sums up the results of 3 and 13 of this chapter and which is valid for an arbitrary space R m , will be found in the note of F. Roger [2]). As usual, the hyperplanes in jK 2 and R$ are termed straight lines and planes respectively. Moreover, in the case of plane sets we shall speak of tangent (intermediate, extreme, unique) in place of tangent straight line (intermediate, extreme, unique). We shall discuss the case of the plane (3) and that of the space ( 13) separately, although the proofs of the fundamental theorems 3.6 and 13.7 which correspond to these two cases, are wholly analogous. The proof of the former is, however, more elementary, whereas the latter requires some subsidiary considerations connected with the notion of total differential (cf. below 12). 264 CHAPTER IX. Derivates of functions of one or two real variables. 3. Fundamental theorems on the contingents of plane sets. For brevity, we shall say that the contingent of a plane set E at a point a is the whole plane, if it includes all half-lines issuing from this point. Similarly the contingent of E at a point a will be said to be a half -plane, if E has at this point an extreme tangent I and if contg/?a consists of all the half -lines issuing from a and situated on one side of I. We shall see in this that, given any plane set E, at each point a of E except at most in a subset of zero length, either lthe contingent of Jis the whole plane, or 2 it is a half -plane, or finally 3 the set E has a unique tangent. This result (together with the more precise result contained in Theorem 3.6) was first stated by A. Kolmogoroff andJ. Vercenko f 1 ;2]. It was rediscovered independently, and generalized to sets situated in any space R m , by F.Roger [2]. The proofs, together with some interesting applications of the theorem of Kolmogoroff and Verienko, will be found in the notes of U. S. Haslam-Jones [2; 3]. (For the first part of Theorem 3.6 cf. also A. S. Besicovitch [4].) A finite function of a real variable F, defined on a linear set E, is said to fulfil the Lipschitz condition on E, if there exists a finite number y such that \F(x 2 )F(x 1 )\^.N'\x 2 x 1 \ whenever x l and x 2 are points of E. As we verify at once, we then have A{B(.F;7?)}< <(A T +1)-|| (for the notation, cf. Chap. II, 8, and Chap. Ill, 10). Thus, if a function F fulfils the Lipschitz condition on a set E of finite [zero] outer measure, its graph E(F-,E) on E is of finite [zero] length. It is also easy to see that any function which fulfils the Lipschitz condition on a linear set E, can be continued outside E so as to fulfil the Lipschitz condition on the whole straight line J?j and so as to be linear on each interval contiguous to E. (3.1) Lemma. Let R be a plane set, a fixed direction and P the set of the points a of R at which contg# a contains no half -line of direction 6. Then (i) the set P is the sum of a sequence of sets of finite length, and (ii) at each point a of P, except at most at those of a subset of length zero, the set R has an extreme tangent such that the side of the tangent containing the half-line ad is its empty side. In the particular case in which Q is the direction of the positive semi- axis of y, the set P is expressible as the sum of an enumerable infinity of sets each of which is the graph of a function on a set on which the function fulfils the Lipschitz condition. [3] Fundamental theorems on the contingents of plane seta, 265 Proof. By changing, if necessary, the coordinate system, we may suppose in both parts of the theorem that is the direction of the positive semi-axis of y. Let us denote, for every positive inte- ger n, by P n the set of the points (x,y) of P such that the inequali- ties \x'x\^\ln and \y'~ 0j<l/u imply y f y<^n-\x' x\ for every point (x',y') of It. Since there is no point a of P at which the contingent of R contains the half-line with the direction of the positive semi-axis of t/, it is clear that P=^P fl . Let us now express each P n as the sum of a sequence {P/i,*U 1,2, of sets with diameters less than 1/w. We shall then have |j/ 2 yJ^w-Lr.,- -jpj for every pair of points, (x { ,y { ) and (# 2 , i/ 2 ), belonging to the same set P,,,*. Let (J,,,* be the orthogonal projection of P,,.* on the axis of x. We easily see that each point of Qn tk is the projection of a single point of P,,,/,. Consequently, the set Pn fk may be considered as the graph of a function JV* on (?> Moreover we havt* \F n ,n(jrt)- F nift (xi)\^n-\Xt x\\ for each pair of points x l and $2 f Qn,k, i.e. the function P /lf/r fulfils the Lipschitz condition on Q ntk and therefore (cf. above p. 264) each set P,,,/, B(F llf/r ; $,*) is of finite length. Thus, since P=^,P n ,^ we obtain the required expres- sion of the set P as the sum of an at most enumerable infinity of sets of finite length, which are at the same time graphs of functions fulfilling the Lipschitz condition on sets situated on the r-axis. It remains to examine the existence of an extreme tangent to the set R at the points of P. For this purpose, let us keep fixed for the moment a pair of positive integers w and fr, and let Q ntft be the set of the points of Q n j which are points of outer density for Q n ^ k and at which the function F,,,* is derivable with respect to the set #> Since the set Qn,kQn,k is of measure zero (cf. Theorem 4.4, Chap. VII) and since the function F ntk fulfils the Lipschitz condition on Q n ,^ it follows that A f B( F n ,*; <2,u-<?*,*)] = 0. We need, therefore, only prove that R has an extreme tangent at each point of the set B^ 1 ,,,/,; <?,/,) and that, further, the side of this tangent which contains a half -line in the direction of the positive semi-axis of y is its empty side. Let (f , ?? ) be any point of B(*V A ;<?,,,*)> and A the derivative of F n ,k at with respect to the set Q n%k . Let e be a positive number legs than 1. Since is a point of outer density for the set <?,*, we can 266 CHAPTER IX. Derivates of functions of one or two real variables. associate with each point (f, 77), sufficiently close to ( , r} ), a point G'eQn.k such that (3.2) |'-f |^|_ | and (3.3) |f'-f|<e-|f (for otherwise, the outer lower density of Qnj, at would not ex- ceed 1 e). Bemembering now that %==F nlt (S Q )j let us write for brevity We shall have (3.4) r ,- % -A g -(f- )=D aill ((') + [fi- F n (?)] + A -(t'-t). Now suppose that the point (,17) belongs to R and that |f- |<l/2w 2 and (17 i;J<l/2*. By (3.3), we have |f |<l/n, and, by (3.2), \F n M (D-1 a \** n.|f-fj<n.|f fj<l/2n, so that |-F..A(n->rt<l/. Since the point (', *V*(f )) belongs to P..*CPi. i* follows from the definition of the set P n that rjF ni i( and using (3.3) again, we derive from (3.4) that Now as , and therefore ', tends to f , the ratio !),,.*(')/( ' ) tends to zero; the same is therefore true, on account of (3.3), of the ratio Dn,k(')l(o)- Consequently, since e is an arbitrary positive number less than 1, it follows from (3.5) that the upper limit of the ratio fa-V- Af( o)]/l <>l> as the P int (^^ R tends to (* >%) is non-positive. Further, since the line y r^^=A Q -(x f ) is plainly an intermediate tangent of the set B(-P,,,*; Q n ,k)CR at the point (f , q ), we see that this line is an extreme tangent of the set R at this point and that the half -plane y 7j^A -(x 1 ), which contains the half -line issuing from (f , ^ ) in the direction of the positive semi-axis of t/, is an empty side of this tangent. This completes the proof. (3.6) Theorem. Given a plane set JK, le* P be a subset of R at no point of which the contingent of R is the whole plane. Then (i) the set P is the sum of an enumerable infinity of sets of finite length and (ii) at every point of P, except at those of a set of length zero, either the set R has a unique tangent or else the contingent of R is a half-plane. [3] Fundamental theorems on the contingents of plane sets. 267 Proof. Let {6 n } be an everywhere dense sequence of directions in the plane and, for each positive integer n, let P n denote the set of the points of P at which the contingent of 7? does not contain the lialf- line of direction O n . We clearly have P=P n , and by the preceding /i lemma each set P a , and therefore the whole set P, is the sum of a se- quence of sets of finite length. Further, the same lemma shows that the set R has an extreme tangent at every point of P, except at most in a set of length zero. Now let Q be the set of the points of P at which 1 there exists an extreme tangent which is not a unique tangent and 2 the con- tingent of R is not a half -plane. For each positive integer n, let Q n denote the set of the points ft of Q such that the half-line bO n is situated on the non-empty side of the extreme tangent of R at ft, but does not belong to contg^ft. Plainly Q=VQ n . Now, by the preceding // lemma, for every point bfQ nJ except at most those of a sot of length zero, the half-line bO n is situated on the empty side of the extreme tangent at ft. It follows that all the sets Q n , and therefore also the whole set #, are of length zero. Hence, at every point of P, except perhaps those of a subset of length zero, either there is a unique tangent or the contingent at this point is a half-plane. (3.7) Theorem. Given a plane set R, let P be a subset of R at every point of which the set R has an extreme tangent parallel to a fijed straight line D. Then the orthogonal projection of P on the line at right angles to D is of linear measure zero. Proof. We may clearly assume that the line D coincides with the axis of x. Let 8 and T denote, respectively, the sets of the points (f, rj) of P for which the half-planes y^rj and y^i] are respectively the empty sides of the extreme tangents. Consider the former of these sets. By Lemma 3.1, the set 8 is the sum of a sequence of the sets B(F n ;#J, where the Q n are sets on the .r-axis and the F n functions fulfilling the Lipschitz condition on these sets, respectively. We may suppose (cf. p. 264) that each function F n is defined, and fulfils the Lipschitz condition, on the whole .r-axis and is linear on the intervals contiguous to the set Q n . This being so, we easily see that, for every n, the relation F^(x)^Q^Fn(x) holds at each point x of Q n which is not an isolated point on any side for Q n , i.e. (cf. 1, p. 260) at all the points of Q n , 268 CHAPTER IX. Derivates of functions of one or two real variables. except at most those of an enumerable set. Thus by Lemma (5.3, Chap. VII, we have \F n [Q n ]\ for every positive integer n, and since the projection of $ on the t/-axis coincides with the sum of the sets F n [Q n ]j this projection is itself of measure zero. By sym- metry, the same is true of the projection of the set T on the {/axis, and this completes the proof. As an immediate corollary, we derive from Theorem 3.6 the following prop- osition : (3.8) Given a plane set R, let P be a subset of R at each point a of which there exists a straight line through a which contains no half -line of contg^a. Then the set R has a unique tangent at all the points of P except at most those of a subset of length zero. This result can be easily extended to the space (cf. F. Roger [2]) us follows: (3.9) (riven a set R in the space /f,, let P be a subset of R at each point a of which there exists a plane through a which contains no half -line of contg^a. Then (i) the set P is the sum of an enumerable infinity of sets of finite length and (ii) the set R has a unique tangent at all the points of P except at most those of a subset of length zero. Proof. Let { n } be an everywhere dense sequence of directions in the space /j. For each positive integer A, let P n h denote the set of the points a of P such -> that |cos(a&, tt n )\>l/h for every point b of R distant less than l/h from a. We express each set P n h as the sum of a sequence {P n h k } k ^ , 2 of sets of diameter less than I/A. We then have Keeping, for the moment, the indices n, h, k fixed, we choose a new system of rectangular coordinates, taking for the positive semi -axis of z the half -line of direction H n . Let , /? and y be, respectively, the three positive semi-axes of the new coordinate-system. For any set, or any point, Q, we denote by Q^ a \ Q^ and Q^ the orthogonal projections of Q on the planes i*y, ya and aft, normal to the axes , ft and y respectively. _^ We have |cos(afc, y)\>\/h whenever aeP nftft , b e R and 0<0(a, 6)<1/A. It follows at once that there is no point P^ htk at which the contingent of the plane set R ( ") contains a half -line at right-angles to the semi-axis y. Hence, by (3.8), the set Pj"i A is the sum of an at most enumerable infinity of sets of finite length, and the set R^ has a unique tangent at all the points of P^ ^ except at most those of a set M n h k of length zero. Similarly, the set R^ has a unique tangent at all the points of P^\ h except at most those of a set N n h k of length zero. It follows that the set R has a unique tangent at each point a of P n h A , except perhaps when o (rt) t M n h k or when o^ c N n h ^ Now we easily see that the two ratios (>(a, 6)/<>(a (rt) , b (n) ) and <>(a, 6)/(>(a (/?) , 6 (t<?) ) remain bounded (by h) when a and 6 belong to the set P n<h ^- It follows that the set of the exceptional points of P nhfk at which the set R has no unique tangent is, with the sets M nhk and N n h k , of length zero. For the same reason, since the set P ( \ k is the sum of an at most enumerable infinity of sets of finite length, so is also the set P,,/,** This com- pletes the proof, on account of the relation (3.10). [4] Denjoy's theorems. 269 4. Denjoy's theorems. We shall apply the results of the preceding to establish certain important relations, valid almost every- where, which connect the Dini derivates of any function what- soever, and which are known by the name of the Denjoy relations. For simplicity of wording, we agree to call opposite derivates of a function F at a point X Q the Dini derivates F*(x Q ) and F~(x Q ), or else F + (x ) and F (x ). We shall begin with some preliminary remarks. Let F be a finite function defined in a neighbourhood J of a point XQ and let B denote the graph of F on J. It is clear that if the function F is derivable at the point # , the set B has at (x Q ,F(x Q )) a unique tangent not parallel to the axis of y. Similarly, if two op- posite derivates of F are finite and equal at # , the set B has at (X O ,F(XQ)) an extreme tangent y F(x ) = k-(x- X Q ], whose angular coefficient k is equal to the common value of these derivates. Con- versely, if at the point (x OJ F(x 9 )) the set B has the extreme tangent y~~ F(x Q ) = k-(x~ .T ) where fc+oo, then 1 F+(XQ)=F~(X O ) = k in the case in which the half-plane y y Q ^k>(x x {} ) is an empty side of this tangent and limsup,F(#)<.F(a? ), and 2 F*(x )=F (x Q )=^J< in x-*x<> the case in which the half -plane y y Q ^k-(x ;r ) is an empty side and limintF(x)^F(x Q ). jr-Mo In the enunciations of the theorems which follow, we shall frequently be concerned with exceptional sets E } connected with a function F and subject to the condition A{R(F; E)}=0. This condition evidently implies both |<E| = and \F[E]\ = Q, since the sets E and F[E] are merely the orthogonal projections of the set B) on the x- and y-axes, respectively. (4.1) Theorem. If at each point of a set E one of the extreme uni- lateral derivates of a function F is finite, this derivate is equal to its opposite derivate at every point of E except perhaps at the points of a set E^ of measure zero such that A{B(F;E l )} = 0. Proof. We may clearly suppose that the same derivate, F*(x) say, is the one which is finite throughout E. We thus have lim sup F(x) < F(x ) at every point x eE and, on account of Theorem 1.1 (ii), we may even suppose that limsup.F(#)<^(#o) at every point x of E. Now, when # el?, the contingent of B(F;E) at the point (x Q ,F(x Q )) contains no half -line situated in the half-plane x^x and having 270 CHAPTER IX. Derivates of functions of one or two real variables. angular coefficient exceeding F^(XQ). Therefore, by Theorem 3.6, the set B(jP; E) has an extreme tangent at each of its points (x ,F(x )) J except for those of a subset B l of length zero, and this tangent has the half-plane y y Q ^F*(x Q )-(x x ) for its empty side. Hence, denot- ing by E l the orthogonal projection of B^ on the#-axis, we see, from the remarks made at the beginning of this , that at every point x of the set E E^ the derivates F*(x) and F "(x) are equal. This com- pletes the proof since A(B(F;E l )}=A(B J ) = 0. (4.2) Theorem. If at each point of a set E a finite function F has either two finite Dini derivates on the same side, or else a finite extreme bilateral derivate (F(x) or F(x)), then the function F is almost every- where derivable in J; moreover, denoting by E Q the set of the points x of E at which the function F is not derivable, we have A(R(F;E )}z=Q. Proof. It will suffice to consider separately the following two cases: 1 The function F has two Dini derivates on the same side finite at each point of E. We then have, by Theorem 4.1, (4.3) F~*~(x)=F~~(x) and F+(x)=F~(x) at each point x of J5, except perhaps those of a set E such that A(B(P;E )} = 0. But the relations (4.3) imply the equality of all four Dini derivates at the point x, and since two of them are finite, by hypothesis, at each point x of E, the function F is derivable through- out EE Q . 2 The function F has an extreme bilateral derivate finite at each point of E. By applying twice over Theo- rem 4.1, and making use of the obvious relations F*(x)^F*(x) and F~~(x)^F~(x), we see that the four Dini derivates are finite and equal at each point of E, except perhaps at those of a set on which the graph of F is of zero length. This completes the proof. Theorem 4.2 (in a slightly less complete form, it is true) has already been mentioned in Chap. VII, p. 236, as a corollary of Theorems 10.1 and 10.5, Chap. VII. We have also stated that (as a consequence of these same theorems) the set of the points at which a function has a unique derivative (even a unilateral derivative) infinite, is necessarily of measure zero. We can now extend this result by taking the modulus, as follows: (4.4) Theorem. For any finite function F, the set of the points x at which lim \F(x+h) F(x)\/h= + oo, is of measure zero. [|4] Denjoy's theorems. 271 Proof. Denoting the set of the points in question by A, we see at once that the graph of the function F has at every point of the set B(F;A), except perhaps at those of a set of length zero, an extreme tangent parallel to the y-axis. Thus, by Theorem 3.7, the set A, which is the projection of the set R(F;A) on the #-axis, is of measure zero and this completes the proof. It results, in particular, from Theorems 4.1, 4.2 and 4.4 that the Dini derivates of any finite function F satisfy one of the following four relations at almost every point x: lOF+(aO=.F "(*)==+ F^(x)==F~(x)= : ~oo; 2F+(x)=:F~(x)^oo t F+(x)=:oo, F~~(x)=+oo, 3 F + (x)=F~-(x)oo; F+(x)=+ oo, F~(x)=oo; 4 F + (x)=F+(x)=F~~(x)F~(x)4 oo- For direct proofs of this theorem, which was established first by Denjoy for continuous functions and then generalized to arbitrary functions, vide: A. Denjoy [1], G. C. Young [2], F. Riesz [7], J. Rid- der [4], J. C. Burkill and U. S. Haslam-Jones [1], and H. Blumberg [2] (cf. also A. N. Singh [1]). A further discussion of the Denjoy relations will be found in the notes of V. Jarnik [1] (for functions of one variable) and of A. S. Besi- covitch [6] and A. J, Ward [4] (for functions of two variables). For Theo- rem 4.4 see S. Saks and A. Zygmund [1] (cf. also S. Banach [1]). A part of the Denjoy relations has recently been generalized to differential coefficients of higher orders; see the important memoirs of A. Denjoy [9], J. Marcinkiewicz and A. Zygmund [1], and J. Marcinkiewicz [2]. We may now supplement Lemma 6.3, Chap. VII, by the fol- lowing result: (4.5) Theorem. Let M be a finite number and F a finite function such that \F+(x)\^M at every point x of a set E. Then \F[E]\ < M-\E\. Proof. Let E l denote the set of the points x of E at which F-(x)=F+(x). By Theorem 4.1, we have AIBCF;^)}^ and there- fore, I-FCJEJ^O. On the other hand, since \F~(x)\=\F^(x)\^M at each point xeEE^ it follows from Lemma 6.3, Chap. VII, that \F[E JSyl < M-\E\, and this completes the proof. An immediate consequence is the following criterion for a func- tion to fulfil Lu sin's condition (N) (Chap. VII, 6): (4.6) Theorem. If a finite function F has at each point x of a set E a finite Dini derivate, the function necessarily fulfils the condition (N) on E. Proof. It is enough to show that if at each point a? of a set H of measure zero the function F has one of its Dini derivates, F* say, finite, then \F[H]\ Q. For this purpose, let H n be the set of the points xeH at which |J^(o?)|<n. We have, by Theorem 4.5, -|H n | = for each positive integer n, and hence |.F[JEn|=0. 272 CHAPTER IX. Derivates of functions of one or two real variables. It is easy to see that the hypotheses of Theorem 4.5 imply that A{B(F; E)}^.(M+ l)-\E\. This remark enables us to complete Theorem 6.5 of Chap. VII, as follows: // the derivate F+ of a finite function F is finite at every point of a measurable set E, except at those of an enumerable subset, then the function F, together with its derivate F^~, is measurable on E and we have \F[E]\* f\F+{x)\dx and A<B(F; E)}= f\l+ [F+(x)?} l/ *dx. E k We may note also the following consequence of Theorem 4.5: // one of the four Dini derivates of a function F vanishes at every point of a set E, then \F[E]\Q. For functions F(x) which are continuous, or more generally continuous in the Darboux sense (i. e. assume in each interval [a, b] all the values between F(a) and F(b)), we deduce at once the following result: (4.7) Theorem. If F is a finite function, continuous in the Darboux sense on an interval I, and if at each point of this interval, except those of an enumerable set, one at least of the four Dini derivates is equal to zero, then the function F is constant on I . *5. Relative derivates. The Denjoy relations can be extended in various ways to relative derivates of a function with respect to another function. Let us remark that, in accordance with the definition given in Chap. IV, p. 108, the extreme derivates of any function with respect to a finite function U are determined at each point which belongs to no interval of constancy of the func- tion 17; consequently, the set of values taken by the function U at the points at which the extreme derivates with respect to U remain indeterminate is at most enumerable. In the sequel it will be useful to employ the notation adopted in Chap. IV, 8. Let us recall in particular, that if C is a curve given by the equations xX (), y = Y(), its graph on a linear set E (i.e. the set of the points (X(t),Y(t)) for t E) is denoted by B((7;JB). (5.1) Lemma. If C is a curve given by the equations x U(t), y = F(t), the set E of the points t at which F u (t)<+oo J may be expressed as the sum of a sequence of sets {E n } such that (V) for every n and for every open interval I of length less than 1/n, the set B(C;I) has a unique tangent at every point of B(C;I-E n ) except those of a set of length zero. Proof. Let us denote, for each positive integer n, by E n the set of the points t such that, provided that the differences F(t')F(t) and U(t') U(t) do not vanish simultaneously, the inequality \t't\^.l/n implies [F(t')F(t)]/[U(t')U(t)]^n. We see at once that, for any [5] Relative derivates. 273 open interval I of length less than 1/n, the contingent of B(<7;/) at a point of B7; /!?) cannot contain half -lines of angular coef- ficient greater than n, and can be, therefore, neither a whole plane, nor a half -plane (cf. 3, p. 264). The property (y) of the sequence {E n } thus appears as a direct consequence of Theorem 3.6. (5.2) Theorem. If U and F are continuous functions and we have Fc(t)<-\-oo at every point t of a set E, then there is a finite derivative F'u(t) at each point t of E, except at the points of a set H such that Proof. Let C denote the curve given by the equations x U(t), y = F(t). On account of Lemma 5.1, the set E is expressible as the sum of a sequence of sets {E n } which fulfil the condition (v) of this lemma. Keeping fixed, for the moment, a positive integer n, let us consider an open interval / of length less than 1/n. Let B n (I) denote the set of the points of B((7;/-JS' / ,) at which the graph of the curve C on / either has no unique tangent, or else has a unique tangent parallel to the y-axis. Further, let B n (I) be the projection of the set B n (I) on the #-axis. On account of the condition (v) and Theorem 3.7, we have |J3 n (/)| = 0. Now since U and F are continuous, it is clear that the derivative F'u(t) exists and is finite at each point tel-Enj provided that U(t) does not belong to the set /?(/). Hence, / being any open interval of length less than 1/n, this derivative exists and is finite at each point teE n , except at most at the points of a set H n such that \U[H n ] =0. This completes the proof, since (5.3) Theorem* If U is a continuous function and F any finite function for which Fu(t) = at each point t of a set E, then \F[E]\ = Q. Proof. Let C denote, as in the proof of the preceding theorem, the curve x= U(t), y=F(t), and let E be expressed as the sum of a sequence of sets {E,,} subject to the condition (v) of Lemma 5.1. Keeping fixed, for a moment, a positive integer n, let us consider any open interval / of length less than 1/n, At each point of B(C; !#), except those of a set of length zero, the set B(C;/) then has a unique tangent, and since the function U is continuous and Fu(t) = Q at each point teE, this tangent is parallel to the #-axis. It follows, by Theorem 3.7, that the set F [/"], which coincides with the projection of the set R(C,I'E n ) on the y-axis, is of measure zero. Since / is any interval of length less than 1/n, it follows that \F[E n ]\=Q for each positive integer n, and finally that \F[E]\ = Q. 274 CHAPTER IX. Derivates of functions of one or two real variables. The hypothesis of continuity of the function U is essential for the validity of Theorems 5.2 and 5.3 (the hypothesis of continuity of the function F, which is not required in Theorem 5.3, may, however, be removed also from Theorem 5.2). Let F(t)~ -t identically, and let U(t) = t for irrational values and U(t)=t+l for rational values of t. Denoting by E the set of irrational points of the interval (0, 1), we shall have at each point t of this set F(*)=0 and Nevertheless | /"[/? l|-=|F[JffJ|=l. On the other hand, the hypothesis of continuity of the function V may be removed from Theorem 5.3, if we replace the con- dition Fy(t)Q by the more restrictive condition Fy(t)to. To see this, we shall first establish an elementary lemma. (5.4) JLenima. // U is a finite function on a set E, there exists a set TCE such that the set U[T] is at most enumerable and such that each point TtE T is the limit of a sequence of points {t t } of E which fulfils the conditions (i) t t >r and U(t,)^-U(r) for each i 1,2,... and (ii) lim U(t,)=U(T). i Proof. Let T be the set of the points rtE none of which is the limit of a sequence {t t } of points of E subject to the conditions (i) and (ii) of the lemma. Let us denote, for each positive integer A, by T k the set of the points T of T for which there is no point t e E such that both 0< r<l/k and 0<\U(t)U(T)\<lfk. We have T=.Tk. Plainly the function U cannot, on any portion of T k of k diameter less than I/A;, assume two distinct values differing by less than I/A;. It follows that each set U[T k ] is at most enumerable, and the same is therefore true of the whole set U[T]. (5.5) Theorem. If U and F are any finite functions and F v (t) = or, more generally Fv(t)Flj(t)~Q, at each point t of a set E, then Proof. Let C be the curve x= U(t), y=F(t) 9 and let E n denote, for each positive integer n, thfi set of the points t of E such that the inequality 0<r *<l/n implies \F(V)F(t)\^\U(t') U(t)\ whatever be the point t'. We can express each set E n as the sum of a sequence {E n ,k}it=-\&.. of sets of diameter less than 1/n. Let us keep n and k fixed for the moment. It is clear that the contingent of the set B(C;E n ,k) cannot, at any point of this set, contain a half -line whose angular coefficient exceeds the num- ber 1 . Consequently, denoting by /?,* the set of the points of B ( C\E ntk ) at which the set ^R(C;E ntk ) has no unique tangent, we see from Theorem 3.6 that A(B n ) = Q. [5] Relative derivates. 275 Now the set E n>k contains, by Lemma 5.4, a subset T n ,* such that U[T n ,k\ is at most enumerable and that each point reEn^ T n>k is the limit of a sequence {</} of points of E n , k which fulfils the con- ditions (i) and (ii) of this lemma. Hence, the relations Fu(t)=F(j(t)=Q being satisfied, by hypothesis, at each point te E n , k , the set B(C; /JiA ) has a unique tangent parallel to the o?-axis at each point of the set R(C;E n , k 2V*) ,*. Since A(B nth ) = Q, it thus follows from Theorem 3.7 that the set F[E n%k T n , k ~\, which coincides with the projection of the set H(C;E nik T,,,*) on the y-axis, is of measure zero. The same is therefore true of the set F[E n ^ for the set F[T nik ] is, with U[T ntft ], at most enumerable. It follows at once that Q, since E= We may mention an application of Theorems 5.3 and 5.5, which is connected with the following theorem of H. Lebesgue [II, p. 299]: If the derivative of a con- tinuous junction F, with respect to a junction U of bounded variation, is identically zero, then the junction F isaconstant. J. Petrovski [1] and R. Caecioppoli [1] extended this theorem, in the case when the function U is continuous, by removing the hypothesis of bounded variation for U. At the same time, Petrovski remarked that it was sufficient for the validity of the theorem to suppose that the relation F'y(t) holds everywhere except in an enumerable set. It is easy to see that this result is contained in each of the separate theorems 5.3 and 5.5. These theorems actually enable us to state the result of Petrovski and Caccioppoli in two slightly more general forms. Thus: 1 Suppose that U and F are continuous junctions and that at each point t, except at most those of an enumerable set, one at least of the jour relations FyW^O, ^^(f)=0, F+(t)=F}j(t)=0 or Fjj(t)=F-(t)^0 is fulfilled. Then the function F is a constant. 2 Suppose that U is any finite function and F a continuous function, and let one of the relations F^(t)^F^ T (t)=Q or F^W^F^t)^^ hold at each point t except at most those of an enumerable set. Then the function F is a constant. We observe further that, in both the statements 1 and 2, we may replace the hypothesis of continuity of F by the hypothesis that F is continuous in the Darboux sense (cf. 4, p. 272); moreover the condition that the exceptional set be at most enumerable may be replaced by the condition that the set of values assumed by the function F at the points of this set be of measure zero. The Denjoy relations have a more complete extension to rela- tive derivates when the function U of Theorem 5.2 is subjected to certain restrictions. Thus: (5.6) Theorem. Let U and F be finite functions, and suppose that, at each point t of a set E, the derivative U'(t) (finite or infinite) exists and that F^(t)<-\-oo. Then Fu(t)=F$(t)=oo at each point t of E except at most the points of a set H such that \U[H]\ 0. 276 CHAPTER IX. Derivates of functions of one or two real variables. Proof. We may clearly restrict ourselves to the case in which the derivative U'(t) is non-negative throughout E, and even, by Theorem 4.5, to the case in which (1) U'(r)>0 at each point r of E. We then have Hmsup J7(XC7(T)^liminf U(t) at each point /-H- f-H-f TfE, and consequently, on account of Theorem 1.1 (ii), we may suppose that (2) the function U is continuous at each point of E. This implies that we then have also at each point r of JE7, and hence, appealing again to Theorem 1.1 (ii), we may suppose further that (3) limsup J F(J)<jF(T) at each point t-n rtE. Finally by Theorem 1.2, we may suppose (4) FU(T)^FV(T) at each point r<-E. Let now (1 be the curve defined by the equations #= U(t), y = F(t). We denote, for each positive integer n, by E n the set of the points it E such that, for every point *', (i) the inequal- ity 0' t<l[n implies the two inequalities U(t')>U(t) and F\t') F(t)<n>[U(t') U(t)], and (ii) the inequality 0<f-f<l/n implies U(t)>U(t f ). Since, by hypothesis, Fv(t)<+oo and since, by (1), U'(t)>0 at each point t of /J, we see that E=E n . n Keeping a positive integer n fixed for the moment, let / be any open interval of length less than 1/w. Whenever (fiy) is a point of R((';l-En)j the contingent of the set B(C;Z) at (f,*?) clearly con- tains no half -line which is situated in the half-plane x^ and which has an angular coefficient exceeding n. Let D(I) denote the set of the points of the set B(C;/) at which this set has an extreme tangent, non-parallel to the y-axis, with an empty side containing the half-line in the direction of the positive semi-axis of y. Further, let B n (l) be the set of the points of B((7;Z- n ) which do not belong to Z>(/), and let B n (I) be the projection of B n (I) on the ,T-axis. By Theorems 3.6 and 3.7, the set B n (I) is of measure zero. This being so, let J be any point of the set I-E n such that U(t ) does not belong to the set B n (I). Let us denote by fc the angular coefficient of the extreme tangent to the set B(C;/) at the point (U(t Q )^F(t^)). It follows easily from the hypotheses (1), (2) and (3) that Fud^k^F^t^^nd. tnis, in view of (4), leads to the relation Fu(t ) --=F(e )4=oo, The Banach conditions (T^ and (T 2 ). 277 Thus, since I is any open interval of length less than 1/n, we find that the last relation holds at each point t of E n other than those belonging to a set H n such that |E7 [#]! = (). This completes the proof, since we have seen that E= In view of Theorem 7.2, Chap. VI I, we derive from Theorem 5.6 the following theorem which has been established in a different way by A. J. Ward [3]: (5.7) Suppose that the function U is VBG* and let F be any finite function. Let E be a set at each point t of which we have either Fy(t)<+ oo or Fy(t)> - Then the derivates Fy and F$ are finite and equal at all points of E except at most those of a set H such that |Z7[J5F]|=0. It will result from the considerations of 6 (see, in particular, Theorem 6.2) that Theorem 5.7 remains valid for all continuous functions U which fulfil the condition (T!). Nevertheless, its conclusion ceases to hold if we allow U to be any function which is VBG or even ACG. To see this, let be a non-negative continuous function which is ACG on the interval [0, 1] and for which G(t)=0 and G~(t)< 1 at any point t of a perfect set E of positive measure (for the construction of such a function cf. Chap. VII, 5, p. 224). Let us choose V(t)=t+ G(t) and F(0=t. We shall then have at jjvery point t of E, U(t)=t, U + (t)^l and ?7~(J)<0<t7 (t), so that ()^F+(t)^F+(t)-^ 1, while Fy(t)=oo and F~(()-+oo. Nevertheless \U[E]\^~.\E\>0. (This example is due to Ward.) *$6. The Banach conditions (TJ and (TJ. A finite func- tion of a real variable F is said to fulfil the condition (T t ) on an interval I if almost every one of its values is assumed at most a finite number of times on I. A finite function F is said to fulfil the condition (T 2 ) on an interval / if almost every one of its values is assumed at most an enumerable infinity of times on /. These two conditions were formulated by S. Banach [6]. We shall begin by studying the condition (TJ and we shall establish a differential property which is equivalent to this condition in the case when F is continuous (vide below Theorem 6.2). Another equivalent condition, due to Nina Bary, will be estab- lished in 8 (Theorem 8.3). (6J) Lemma. Suppose that F is a continuous function and that E is a set at no point of which the function F has a derivative (finite or infinite). Suppose further that each point x of E is an isolated point of the set E[F(t) = F(x)]. Then A(B(F;E)) = Q, and come- t quently |JB| = \F[E]\ = 0. 278 CHAPTER IX.. Derivates of functions of one or two real variables. Proof. For each xeE there exists a neighbourhood / such that, when tel, the difference F(t) F(x) remains of constant sign as long as t remains on the same side of x\ this difference then changes sign as t passes from one side of x to the other, except in the case in which the function F assumes a strict maximum or mini- mum at x. Therefore, if we denote by E the set of the points at which the function assumes a strict maximum or minimum, we see at once that the four Dini derivates of F have the same sign at any point ,r of E E Q . In other words, since, by hypothesis, the function F has no finite or infinite derivative at any point of E, we shall have at each point x of EE Q either + oo>F(x)^Q or else oo<F(x)^Q. Hence, by Theorem 1.2, A (B(F;#-E )} =_-<), and, since the set E Q is at most enumerable (cf. Theorem 1.1), it follows that A(BCF;#)) = 0. (6.2) Theorem. In order that a function F which is continuous on an interval I, fulfil the condition ( r l\) on this interval, it is neces- sary and sufficient that the set of the values assumed by F at the points at ichich the function has no derivative (finite or infinite) be of meas- ure zero. Proof. Denoting by Y the set of the values assumed an in- finity of times by the function F on /, and denoting by E the set of the points of / at which F has no derivative, we have to prove that the relations r=0 and ,F[B]| = are equivalent. 1 Y -=0 implies F[#]j = 0. Let X be the set of the points XL I such that F(x)*Y. Then F[X]=Y, whence \F[X]\-=0. On the other hand, for each x t E JT, the set of the points x such that F(jr) =-F(x ), is finite, and consequently an isolated set. It follows from Lemma 6.1 that \F[E A']| = 0, and hence finally that 2 t F[E\^Q implied |l'i = 0. Let H denote the set of the points x at which F'(x) ^0. By Theorem 4.5, we have |F[/7]| = 0. Now let ?/ he any point of Y F[E], and let /? denote the set of the points x tit which f 1 (x) = y . The set 7? being infinite and closed, let J* be a point of accumulation of E n . Since the function F has ;i derivative at each point of /? , we find that F'(x )~ 0; thus x fH and therefore y fF[H\. It follows that YF[E]Cf[H]i and hence that \YF[E]\ = 0. Thus \F[E]\ = Q implies |Y| = and the proof is complete. [6] The Banach conditions (T x ) and (T a ). 279 (6.3) Theorem. 1 A continuous junction which is VBG* (in par- ticular, one of bounded variation) on an interval I, necessarily fulfils the condition (T x ) on I. 2 A continuous function which is VBG on an interval I neces- sarily fulfils the condition (T 2 ) on I. Proof. On account of Theorem 7.2, Chap. VII, the first part of the theorem is an immediate corollary of Theorem 6.2. To establish 2, let us suppose that F is continuous and VBG on an in- terval I. The interval 1 is then expressible as the sum of a sequence {E n } of closed sets on each of which the function F is VB. We may clearly suppose that each E n contains the end-points of the interval I. Let us denote, for each positive integer n, by F n , the function which is equal to F on E n and which is linear on the intervals contiguous to E n The functions F n are plainly of bounded variation on I, and therefore, by 1, they fulfil the condition (T t ). It follows at once that the function F fulfils the condition (T 2 ) on /. In the part of Theorem 6.3(1) that applies to functions which ire VBG*, the continuity hypothesis for the function F is not a superfluous one (thus, the function F(x)=-sin(\/x) for x * and F(0)=0 is VBG* and does not fulfil the condi- tion (T,) on [0, 1]). This hypothesis may however be replaced by a weaker one, which consists in supposing that the function F has no points of discontinuity other than of the first kind (i.e. that, at each point x, both the unilateral limits F(x+) and F(x ) exist). In particular, functions of bounded variation, whether continuous or not, all fulfil the condition (Tj) (and from this it follows easily that the continuity hypothesis may be removed altogether from the second part (2) of the theorem). For functions of bounded variation, the condition (T,) may also be deduced from the following general property of plane sets, established by VV. Gross [1] (cf. J. Gillis [1 ]): If E is a plane set and E n denotes the set of the values of r { such that the line yr] contains at least n distinct points of the set E, then A(Ey?n*\E n \. In connection with part 2 of Theorem 6.3, it may be noted further that functions which are VBG, or even A('G, need not fulfil the condition (T,). An example ia furnished by the function U considered in 5, p. 277. The latter is also, SLR will follow from results to be established in 7 (cf. in particular, Theorem 7.4), an example of a continuous function which is ACG, and consequently fulfils the condition (N), without fulfilling the condition (S) of Banach. For continuous functions of bounded variation, the condition (TJ is also a consequence of the following theorem of S. Banach [5] (cf. also N. Bary [3, p. 631]), which contains at the same time an important criterion for a continuous function to be of bounded variation: 280 CHAPTER IX. Derivates of functions of one or two real variables. (6.4) Theorem. Let F be a continuous junction on an interval / =[a,6] and let s(y) denote for each y the number (finite or infinite) of the points of I at which F assumes the value y. Then the function s(y) is measurable (iB) and we have +00 (6.5) Pro of. For each positive integer n,let us put I\ n} [a,a+(ba)l2 n ] and I ( k n} =(a + (k-l)(b-a)!2",a + k(b-a)l2 n ] j when fc=2,3,...,2". This defines a subdivision 3 (/l) of the interval / into 2" subintervals, of which the first is closed and the others are half-open on the left. For i=l,U,...,2 n , let s* l} denote the characteristic function of the set F[I ( f } ], and let s (n \y)= ^(y). h^\ We see at once that the functions s (n) (y) constitute a non- decreasii/g sequence which converges at each point y to s(y). Hence, the functions ,v (/0 (//) being clearly measurable (33), so is also the func- tion s(y). +00 On the other hand, l's^ } (y)dy = \F[I ( ^]\=O(F^I ( ^). Therefore, 00 denoting by W (n the sum of the oscillations of the function F on the -l-oo intervals of the subdivision 3 (/I) , we obtain fs (n \y) = W ( ", and the - -oo relation (6.5) follows by making n->oo. (6.6) Theorem. If F(x) is a continuous function which fulfils the condition (T 2 ) on an interval / , the set I) of the points at which the derivative F'(x) (finite or infinite) exists, is non-enumerably infinite. Moreover, if we write and N then, for each interval / = [a,&]CA> we (-7) -\F[Nl\^F(b)-F( Proof. We may, plainly, suppose that since the other case may be discussed by changing the sign of the function F. Let Y be the set of those values of F on / which are assumed by the function F at most an enumerable number of times on /. Denoting, for each y, by E n the set of the points xel such that [6] The Banach conditions (Tj) and (T t ). 281 ) y, we shall show that with each point ye Y we can associate a point x y eEy, in such a manner that (i) F(x y )^Q and (ii) x !f is an isolated point of the set E y . For this purpose, we remark first that if the set E y redwes to a single point, the latter may be chosen for our x y . For, in that case, the condition (ii) is clearly fulfilled, while the condition (i) holds on account of the hypothesis (6.8). Let us therefore consider the other case, in which the set E t) contains more than one point. Then, since the function F is con- tinuous by hypothesis and j/eY, the set E y is closed, and at most enumerable. This set, therefore, contains a pair of isolated points ,/J between which it has no further points. (This is obvious, if the set E H is finite. If E y is infinite, its derived set (cf. Chap. II, p. 40) is itself closed, non-empty, and at most enumerable; the latter, therefore, contains an isolated point a? . Thus near X Q there are only isolated points of E y . It will, therefore, suffice to choose, among the latter, any two consecutive points as our points a and ft.) Consequently, at one at least of the points a and /?, the upper deri- vate of F(x) is non-negative. We choose this point as our x y . We then see at once that the conditions (i) and (ii) are fulfilled. This being established, let X denote the set of all the points x y which are thus associated with the points y ^ Y. It follows from the conditions (i) and (ii) and from Lemma 6.1, that \F[XP]\ = \F[X D]| = 0, and so, by definition of the set A', that \Y\ = \F[X]\ = \F[X-P]\^\F[P]. Since the condition (T 2 ) im- plies that |.F[/]| = |Y|, we obtain, in view of (6.8), the inequality -\F[N]\^O^F(b) F(a)^\F[I]\^\F[P]\, i.e. the inequality (0.7). Finally, since this relation holds for every subinterval [a, b] of / , we see that, unless the function F is a constant, one at least of the sets F[N] and F[P] is of positive measure. The set D=X+P is thus non-enumerably infinite, and this completes the proof. (6.9) Theorem. Let F be a continuous function which fulfils the condition (T 2 ) and let g be a finite summable function. Suppose further that F'(x)^.g(x) at each point x at which the derivative F'(x) exists, except perhaps those of an enumerable set or, more generally, those of a set E such that |.F[J5f]| = 0. Then the junction F is of bounded vari- ation and, for each interval [a,&], we have & (6.10) F(b)F(a)^fF'(x)dx. 282 CHAPTER IX. Derivates of functions of one or two real variables. Proof. Let P be the set of the points x of [a, ft] at which the derivative F'(x) exists and is non-negative. Then, since at each point x e P E we have O^.F'(x)^g(x)< + oo J it follows from Theorem 6.5, b Chap. VII, that \F[PE]\^fF'(x)dx^f\g(x)\dx. On the other Pa hand, by hypothesis, \F[E]\ = Q. Hence, on account of Theorem 6.6, b F(b)F(a)^l\g(x)\dx for each interval [a,ft], and, in corise- d quence, F is a function of bounded variation whose function of singularities is monotone non-increasing. The inequality (6.10) fol- lows at once. In view of Theorem 6.3(2), we may apply Theorem 6.9, in particular, to continuous functions F which are VBG. We also observe that Theorem 6.9, when F is of bounded variation, may be deduced from de la Valle'e Poussin's Decomposition Theorem (Chap. IV, 9). Theorem 6.9 may be generalized further, by replacing the condition that the function g is summable, by the condition that the latter is ^-integrabie (the function F then shows itself to be VBG*). We thus obtain a proposition similar to Theorem 7.3, Chap. VI. The proof of Theorem 6.9 in this generalized form is, however, more complicated. * * 7. Three theorems of Banach. We have repeatedly emphasized the importance of Lusin's condition (N) in the theory of the Denjoy integrals. We shall show in this , that every con- tinuous function which fulfils the condition (N), also fulfils the con- dition (T 2 ). This result due to S. Banach [6] (cf. also N. Bary [3, p. 195]) renders Theorems 6.6 and 6.9 applicable to functions which fulfil the condition (N). We shall also study another condition, introduced by 8. Banach [6] and termed condition (S). We say that a finite function F fulfils the condition (S) on an interval / , if to each number c>0 there corresponds an ?j>0 such that, for each measurable set JEC/o, the inequality \E\<ij implies \F[E]\<e. (This condition is essentially more restrictive than the condition (N); cf. the remarks, p. 279, also G. Fichtenholz [4].) (7.1) Lemma. Given a function F which is continuous on am, in- terval I, every closed set ECl contains a measurable set A on which the function F assumes each value yeF[E] exactly once. [7] Three theorems of Banach. 283 Proof. With each yeF[E] we associate the lower bound x y of the set of the points x of E at which F(x) = y, and we denote by A the set of all the points x y which correspond in this way to the values y e F[E]. Since the set E is closed, we plainly have ACE and F assumes on A each of the values yeF[E] exactly once. In order to establish the meaaurability of A, let us denote, for each positive integer w, by E nj the set of the points x E such that E contains at least one point t which is subject to the conditions F(t) = F(x) and xt^lfn. We have A = E%E n , where E is closed n by hypothesis, and where each E n is closed by continuity of F. The set A is thus measurable and this completes the proof. (7.2) Lemma. Let F be a continuous junction which fulfils the con- dition (N) on an interval I. Then (i) every measurable set ECI contains^ for each e>0, a meas- urable subset Q, such that \F[E] F[Q]\<ej and on which the func- tion F assumes each of its values at most once, (ii) every measurable set ECI contains a measurable subset R, such that \F[E]F[R]\ = Q, and on which the function F assumes each of its values at most an enumerable infinity of times. Proof, re (i). As a measurable set, E is the sum of a set H of measure zero and an ascending sequence of closed sets {13,,}. Since the function F fulfils the condition (N), we have \F[H]\ = Q, and hence, the sets F[E n ] being measurable, there exists a positive in- teger n such that \F[E]F[E no ]\<e. Now, by Lemma 7.1, there exists a closed set QCE^ such that each value yeF[E ni ] is as- sumed exactly once by F on Q. This set Q plainly fulfils the condi- tions stated. re (ii). In view of (i), there exists for each positive integer n a measurable set Q n CE, such that \F[E]F[Qn]\<llnj and on which the function F assumes each of its values at most once. Therefore, writing R=2>Q nj we see immediately that \F[E] F[R] \ = and n that on 12 the function F assumes each of its values at most an enumerable infinity of times. This completes the proof. We shall establish in this three theorems due to Banach on functions which fulfil the conditions (N) or (8). The first of these theorems, which concerns functions fulfilling the condition (N), is as follows: 284 CHAPTER IX. Derivates of functions of one or two real variables. (7.3) Theorem. Any continuous function F which fulfils the condition (N) on an interval /, necessarily fulfils also the condition (T 2 ) on I. Proof. Let us denote, for each measurable set EC I, by iR/? the class of all measurable sets RCE which are subject to the following two conditions: (i) \F [E] F[R]\ = Q, and (ii) each value yeF[E] is assumed by the function F at most enumerably often on It. By Lemma 7.2, the class 9t/$- is non-empty, however we choose the measurable set ECI. We shall denote, for any such set Ej by p K the upper bound of the measures of the sets (%?) Consider, in particular, a sequence {//} of sets (9t/) such that \im\H n \ = n r Let H=^H n and let U be a set (ft/..*). We verify /i /i at once that |{7|=0, whence on account of the condition (N), \F[U]\ = (). Therefore |.F[I H] \ =\F[D]\ = 0, so that almost every value y e F[I] is assumed by F only on the set H , and therefore at most enumerably often. The second of the theorems of Banach concerns functions which fulfil the condition (8). (7.4) Theorem. In order that a continuous function F be subject to the condition (S) on an interval /, it is necessary and sufficient that F be subject on I to both the conditions (N) and (T t ). Proof. 1 Suppose that the function F fulfils the condition (S) on /. Since this condition clearly implies the con- dition (N), we need only prove that F fulfils the condition (T^. Suppose then, if possible, that the set of the values assumed infinitely often on / by the function F, is of positive outer measure. Since this set is measurable by Theorem 6.4, it contains a closed subset Y of positive measure. Let X denote the set of all the points x e I such that F(x) e Y. The set Jf, plainly, is also closed. We shall now define by induction a sequence of measurable sets {X,} subject to the following conditions: (i) XrXj = whenever f=H, (ii) IFlX^lYl^ for i=l,2,..., and (iii) the function F assumes each of its values at most once on each set X f . For this purpose, suppose defined the first k sets Xi for which * the conditions (i), (ii) and (iii) are satisfied. Let *=! X f . Since, * /=-.i on Xi, the function F assumes each of its values at most a finite / -- 1 number of times, it follows that each value y e Y is necessarily assumed on the set 15*. By Lemma 7.2, this set therefore contains [7] Three theorems of Banach. 285 measurable subset X*+i such that \F[X^i]\^\F[E k ]\f2^\Y\l2 and that each value is assumed by JP at most once on Xk+\. The sets X\j X%j ...,-XjH-i clearly fulfil the conditions (i), (ii) and (iii). The sequence {X f } being thus defined, it follows from (i) that lim|X/| = 0, and hence, remembering that the function I* 7 fulfils / the condition (S), we have also lim |F[J/]|=0. But this clearly i contradicts (ii), since |Y|>0. 2 Suppose now that the function F fulfils the conditions (N) and (T t ), but not the condition (S). We could then determine a po- sitive number a and a sequence of sets (Ek\ in / so that for k= 1, 2, ..., (7.5) |JS?*|<l/2*, and (7.0) \F[E*]\>o. Let us write E lim sup Ek and A lim sup F[Ek]. We easily see /? * that, if y e A, then either y F[E], or else the value y is assumed by JP on I infinitely often (in fact there exists an increasing sequence of positive integers {&/} and a sequence of points {#/}, every two of which are distinct, such that x { E^ and F(x t ) = y for f = 1,2,...)- Now, on account of (7.5), we have |E| = 0, and therefore also \F[E]\ = Q. On the other hand, by (7.6), |A|>r>0. Thus \AF[E]\^(tj and since, as we have just seen, each value y A ~F[E] is assumed by JP infinitely often on /, this contradicts the hypothesis that the function F fulfils the condition (T a ). We shall establish next a ''differentiability theorem" for the functions which fulfil the condition (N): (7.7) Theorem. In order that a continuous function F be absolutely continuous on an interval / , it is necessary and sufficient that the function F fulfil simultaneously the condition (N) and the condition (7.8) where P denotes the set of the points at which the function F has a finite non-negative derivative. Proof. Since the conditions of the theorem are obviously necessary (cf. Theorem 6. 7, Chap. VII), let us suppose that the function F fulfils the condition (N) and the inequality (7.8). Let g be the function equal to jP'(#) for x P and to elsewhere. Then, if E denotes the set of the points x at which F'(x) = + oo, we shall have F'(x)^g(x) at every point x of I Q E at which the derivative F'(x) exists. 286 CHAPTER IX. Derivates of functions of one or two real variables. On the other hand, since |U| = (cf. Theorem 4.4, or Chap. VII, 10, p. 236), we have |.y[l!f]| = 0, and, since the func- tion F fulfils, by Theorem 7.3, the condition (T 2 ), it follows from Theorem 6.9 that F is of bounded variation on I . This completes the proof, since, by Theorem 6. 7, Chap. VII, every continuous function of bounded variation, which fulfils the condition (N) is absolutely continuous. Theorem 7.7 (in a slightly less general form) was first proved by N. Bary [2; 3, p. 199]. It shows in particular that every continuous function F(x), which is subject to the condition (N) and whose derivative is non-negative at almost every point where F(x) is derivable, is monotone non -decreasing. This proposition contains an essential generalization of Theorem 6.2, Chap. VII. Theorem 7.7 may, moreover, be generalized still further. // a continuous function F(x) fulfils the condition (N) and if the function g(x), equal to F'(x) wherever F(x) is derivable and to elsewhere, has a major function (in the Perron sense), then the function F(x) is ACG* i. e. an indefinite ^-integral. For the part played by the conditions (N), (T,) and (T 2 ) in the theory of Denjoy integrals, cf. also J. Ridder [8]. From Theorem 7.7 we obtain the third theorem of Banach: (7.9) Theorem. Any function which is continuous and subject to the condition (N) on an interval, is derivable at every point of a set of positive measure. * 8. Superpositions of absolutely continuous functions. Suppose given a bounded function G on an interval [a, ft], and a func- tion H defined on the interval [a,/?] where a and ft denote respectively the lower and the upper bound of G on [a, ft]. We call superposition of the functions G and H on [a, ft], the function H(G(x)). The func- tion G is termed inner function and the function H outer function of this superposition. If a function F is continuous and increasing on an interval [a, ft], the continuous increasing function G defined on the interval [F(a).F(b)] so as to satisfy the identity G(F(x)) = x on [a, ft], will, as usual, be termed inverse function of F and denoted by F~ l . It has long been known that the superposition of two abso- lutely continuous functions is not, in general, an absolutely con- tinuous function. By means of the conditions discussed in the preceding , particularly the condition (8), Nina Bary and D. Men- choff succeeded in characterizing completely the class of functions expressible as superpositions of absolutely continuous functions. (Cf. also G. Fichtenholz [3].) [ 8] Superpositions of absolutely continuous functions. 287 (8.1) Theorem. Any function F which is continuous and subject to the condition (T t ) on an interval [a, ft] is expressible on this interval as a superposition of two continuous functions, of which the inner function is of bounded variation and the outer function is increasing and absolutely continuous. //, further, the function F fulfils the condition (N), the inner function of this superposition is necessarily absolutely continuous also. Proof. Let a and /? denote respectively the lower and upper bounds of F on [a, ft], and let Sf(y) denote, for each v y, the number (finite or infinite) of the points of the interval [a, ft] at which F assumes the value y. Since, by hypothesis, the function F is contin- uous and subject to the condition (Tj), we shall have 0<l/s F (y)^l for almost all the values y of the interval [a,/?]. Let us denote by U an indefinite integral of the function which is equal to l/Sf(y) on [a,/?] and to 1 elsewhere. We now write G(x) U[F(x)] for #e[a,ft]. We thus have F(x)=U l [G(x)], and in order to establish the first part of the theorem, it is enough to show that (i) the function f/" 1 is absolutely continuous and (ii) the function G is of bounded variation. Suppose, if possible, that the function U~ } (which is continuous and increasing together with U) is not absolutely continuous. Then (cf. Theorem 6. 7, Chap. VII), there exists a set E of measure zero such that | U~*[E] \ >0. Writing Q=- U~ l [E], we thus have (8.2) |#|>0 and \U[Q]\ = 0. We may, plainly, suppose that the set J?, and therefore the set <?, are sets (CM- Thus (cf. Theorem 13.3, Chap. Ill) \U[Q]\= fV'(y)dy, Q which renders the relations (8.2) contradictory, since almost every- where U'(y)=l/s F (y) >0 for ye [a,/*] and U'(y)=l outside the in- terval [a, /?]. In order to establish (ii), we shall make use of the criterion of Theorem 6.4. Denote for each z by 8 G (z) the number of the points of the interval [a, ft] at which the function G assumes the value z. Since the function D is increasing, we clearly have 8o(U(y)) = SF(y) for each y, and *0(z) = for each z outside the interval [U (a), U(p)]. Hence, remembering that the function V is absolutely continuous, we obtain (cf. Theorem 15.1, Chap. I) 288 CHAPTER XI. Derivates of functions of one or two real variables. + 00 U(fi =fsG U(<*) which shows, by Theorem 6.4, that the function is of bounded variation. Finally, if the given function F fulfils the condition (N), so does the function G(x) = U(F(x)), and the latter, since it is of bounded variation, is absolutely continuous by Theorem 6.7, Chap.VII. This completes the proof. (8.3) Theorem-. 1 In order that a continuous function F be expres- sible as a superposition of two continuous functions of which the inner function is of bounded variation and the outer function is absolutely continuous, it is necessary and sufficient that F fulfil the condition (T T ). 2 In order that a continuous function be representable as a super- position of two absolutely continuous functions, it is necessary and sufficient that the function fulfil both the conditions (T x ) and (N), or what amounts to the same, the condition (S). Proof. Since it follows at once from Theorem 8.1 that these conditions are sufficient, we need only prove them necessary. Let therefore F(x) = H(G(x)) on an interval \a,b], where 6 is a function of bounded variation and H an absolutely continuous function. Let a and ft be respectively the lower and the upper bound of 6 on [a,b]. Let EO and EH denote the sets of the values which the functions G and H assume infinitely often on the intervals [a, b] and [a,/9], respectively. Since the functions G and H fulfil the con- dition (TJ, we have |U0|=|13//|=0, and since the function H is, moreover, absolutely continuous, we have also \H[Eo]\ = Q. Now we see at once that each value which is assumed infinitely often on [a, b] by the function F, belongs either to E Hj or to H[Eo]. The set of these values is thus of measure zero, and the function F ful- fils the condition (T,). If, further, the function G is absolutely continuous (as well as H), then the function F is a superposition of two functions which fulfil the condition (N), and, thei fore, itself fulfils this condition. This completes the proof. [ 8] Superpositions of absolutely continuous functions. 289 Theorems 8.1 and 8.3 are due to Nina Bary [1; 3, pp. 208, 633] (cf. also S. Banach and 8. Saks [1]). Part 2 of Theorem 8.3 was established a little earlier in a note of N. Bary and D. Menchof f [1] (cf. also N. Bary [3, p. 203]) in a form analogous to Theorem 6.2. Thus: (8.4) Theorem. In order that a function F which is continuous on an interval [a,fe] be on this interval a superposition of two absolutely continuous functions, it is necessary and sufficient that the set of the values assumed by the function F at the points of [a<b] at which F is not derivable, be of measure zero. Proof. Let Q F he the set of the points of [a, 6] at which F is not derivable. Suppose first that 1 F(x)H(G(x)) on [a, 6], where H and G are absolutely continuous functions. Let Q G and Q H be respectively the sets of the points at which the functions G and H are not derivable. We have \Q G \=\Q H \ =0 and, consequently, \F[Q G ]\^\H[Q H ]\ = ^ Now, we see at once that if the function F is not derivable at a point x, then either xtQ^ or G(x)eQ H . Therefore F[Q F ]CF[Q G ]+ H[Q H ], and hence, \F[Q F ]\=^0. Conversely, suppose that 2 |jF[0jrl| = 0. By Theorem 6.2, the function then fulfils the condition (T t ). To show that F also fulfils the condition (N), con- sider any set of measure zero, E say, in [a, b]. Since the function I* 1 is derivable at each point of E Q F , we have, by Theorem 6.5, Chap. VII, \F[EQ F ]\=Q, and since, by hypothesis, \F[Q F ]\ = Q, we obtain |-F[/?]| 0. The function F thus fulfils both the conditions (TJ and (N), and is, therefore, by Theorem 8.3 (2), a superposition of two absolutely continuous functions. It follows from Theorem 8.3 (2) that a superposition of any finite number of absolutely continuous functions is expressible as a superposition of two absolutely continuous functions. For the superposition of any finite number of functions which fulfil the condition (S), itself fulfils this condition. The results exposed in this have been the starting point of the important researches of Nina Bary [3] on the representation of continuous functions by means of superpositions of absolutely continuous functions. Let us cite two of her fundamental theorems: 1 Every continuous function is the um of three super- positions of absolutely continuous functions, ami there ejrist continuous functions which cannot be expressed as the sum of two such mi per positions. 2 Kwru continuous function which fulfils the condition (N) or, more generally, wery continuous function which is derivable at every point of a set which has positive measure in each interval is the sum of two superpositions of absolutely continuous functions, and there esist continuous functions which fulfil the condition (X), but are twt expressible as one superposition of absolutely continuous functions (the function U(x) discussed above 290 CHAPTER IX. Derivates of functions of one or two real variables. 9. The condition (D). We shall now establish, for the extreme approximate derivates, a theorem, analogous to Theo- rem 4.6, but whose proof depends on a different idea. It is con- venient to formulate it, from the beginning, in a slightly more ge- neral manner. Given two positive numbers N and s, we shall say that a func- tion F fulfils at a point X Q the condition (D#, f ), if there exist positive numbers ft, as small as we please, such that the difference between the outer measure of the set E[jF(#) F(x Q ) ^ ^ ' ( x X Q)I X and that of the set E[F(x) F(x Q )^ N-(x # ) JC exceeds the number he in absolute value. By symmetry, merely replacing F(x) F(x Q ) and X~-X Q by F(x Q )F(x) and X Q X respec- tively, we define the condition (D^). If, for a point X Q , there exists a pair of finite positive numbers N and e such that the function F fulfils at this point the condition (Djv, f ), or the condition (D^, f ), we say that F fulfils at X Q the condition (D). For measurable functions the condition (D) may be formulated more simply: a measurable function F fulfils at a point x , the con- dition (D), if there exists a finite positive number N such that # is not a point of dispersion for the set of the points x at which (9.1) Theorem. If any one of the four approximate extreme derivates of a function F is finite at a point # , then the function fulfils the con- dition (D) at this point. Proof. Suppose, to fix the ideas, that |JF^,(# )|< -foo and write N=\F^>(x )\+l. Let E l and E 2 be the sets of the points x which are situated on the right of the point # and which fulfil respectively the inequalities F(x)F(x Q )2*N-(xXt) and F(x)F(x^--N-(xXt). It follows at once from the definitions of approximate derivates (Chap. VII, 3) that X Q is a point of dispersion for the set A\, while E 2 has at x a positive upper outer density. Denoting the latter by d, we see at once that the function F fulfils at X Q the condition (DjSf,c) whatever be the positive number (9.2) Lemma. Let N and e be finite positive numbers and suppose that a finite function F fulfils the Condition (D]v,f) at each point of a set E. Then \F[E]\^(2Nle)-\E\. IS 9] The condition (D). 291 Proof. We shall first show that for every interval / = [a,6], (9.3) For this purpose, we write, for every t/, (9.4) The function H thus defined is non-increasing and bounded on the whole/ straight line ( oo, +00); we have, in fact, for every y. (9.5) 0<T(K|I|. Given an arbitrary point y of F[E-I], which is distinct from F(b) and at which the function H is derivable, let us consider a point x^E-1 such that F(x Q ) = y Q . Plainly x Q =b. Let us write, for brevity, and For every subinterval [x QJ x Q +h] of 7, we then have the re- lation B(h,N)CA(h,N)CA(h,~ N)CB(h, N), whence it follows easily, on account of (9.4), that tf(# -#/0-#0/o+^)>TO^ Now, since F fulfils, by hypothesis, the condition (D,v,*) at X Q , there exist positive values A, as small as we please, such that and therefore H(y Nh) H(y + Nh)^he. Hence, for every point y ^=F(b) of F[E-I] at which the function // is derivable. Therefore, denoting, for each positive integer /?, by Q n the part of the set F[E-I] contained in the interval [ w, w], wt find, on account of (9.5), \l\^\H(n)-~ H(~ n)\^e-\Q,,\/2N, from which the inequality (9.3) follows by making n->^o. This being established, let r\ be a positive number and {!*} a sequence of intervals such that (9.6) EC Elk and \E\ + *i^2\lk\. k ft Since (9.3) holds for each interval /, it follows from (9.6) that W+ n ^ (*/2A T ) -Z\F[E !*]!> (sj2N)^\F[E]\ 1 whence, remembering that YI is an arbitrary positive number, we see that \F[E\\<^(2N lf)-\K\. 292 CHAPTEB IX. Derivates of functions of one or two real variables. (9.7) Theorem. If at each point of a set E, a finite function F ful- fils the condition (D) (and so, in particular, if at each point of E the function F has one of its extreme approximate derivates finite), then the function F fulfils the condition (N) on E. Proof. Let H be any subset of E of measure zero, and let H n denote, for each positive integer n, the set of the points x of H at which the function F fulfils the condition (D^i/,) or (D^i,,). We clearly have H=H n , and since, by Lemma 9.2, \F[H n ]\^n 2 -\H n \ = Qj we obtain \F[H]\=Q. Theorem 9.7 enables us to complete Theorems 10.5 and 30.14 of Chap. VII, as follows: (9.8) Theorem. 1 Every finite function F which is continuous on a closed set E and which has at each point of E, except perhaps those of an enumerable subset, either two finite Dini derivates on the same side, or one finite extreme bilateral derivate, is ACG* on E. 2 Every finite function F which is continuous on a closed set E and which has at each point of E, except perhaps those of an enumerable subset, either one finite Dini derivate, or one finite extreme approximate bilateral derirate, or finally two finite extreme approximate unilateral derivates on the same side, is ACG on E. Proof. By Theorems 10.1, 10.5, 10.8 and 10.14 of Chap. VII, the function F is VBG, on E in cavse 1 and VBG on E in case 2. On the other hand, by Theorems 4.6 and 9.7, this function fulfils, in both cases, the condition (N) on E. Hence, by Theorems 6.8 and 8.8 of Chap. VII, the function is ACG, on E in case 1, and ACG on E in case 2. In the most important case in which the closed set E is an interval, Theorem 9.8 may further be stated in terms of Den joy integrals. For this purpose*, let us begin by rioting the following proposition (cf. A. S. Besicovitch [2], and J. C. Burkill and IT. S. Haslam-Jones [1]): (9.9) Theorem. If a finite function F is measurable on a set K and has at each point of this set one of its Dini derivates finite, then this tier irate /.v, at almost all points of E, an approximate derivative of F. Proof. It follows from Theorem 10.8, Chap. VII, that the function F is VBG on E, and so, approximately derivable at almost ;ill th<* points of E. Let us denote by E l the set of the points of K [9] The condition (D). 293 at which one at least of the opposite Dini derivates F + and F~~ is finite. Plainly, F~(x)^F'^(x)^F + (x) at each point x at which the approximate derivative F'^(x) exists, and therefore by Theorem 4.1, F~(x) = F*(x) F 3p (x) at almost all points x of JB 1 . Similarly, we show that F~*~(x)=F~(x)=F f gi v(x) at almost all the points x of E at which one of the derivates F* and F~ is finite. This completes the proof. (9.10) Theorem. 1 If f is a finite function which, at each point of an interval I , except those of an enumerable set, is equal to an extreme bilateral derivate of a continuous function F, then the function f is 5^-in- tegrable on I and the function F is an indefinite 9 ^-integral of f. 2 // / is a finite function which, at each point of an interval I Q , except those of an enumerable set, is equal either to a Dini derivate, or to an extreme approximate bilateral derivate of a continuous function F, then the function f is -integrable on I and the function F is an in- definite ^-integral of f. Proof. In view of Theorem 9.8, the function F is AOG* in case 1, and ACG in case 2. Moreover, at almost all the points x of E, we have F'(x)=f(x) in case 1, and by Theorem 9.9, JFap(jr)- /(&) in case 2. This proves the theorem. Although Theorem 9.8 presents a formal analogy with Theorems 10.5 and 10.14 of Chap. VII, there is an essential difference between the result of this and those of 10, Chap. VII. We see, in the first place, that the criteria of Theorems 10.5 and 10.14 of Chap. VII concern functions which are given on quite arbitrary sets, whereas those of Theorem 9.8 are established only for closed sets. In the second place, if the derivates of a quite arbitrary function satisfy on a set E the conditions of Theorem 10.5, or of Theorem 10.14, of Chap. VII, then the set E can, by these theorems, be decomposed into a sequence of sets on which the function is absolutely continuous. On the contrary, Theorem 9.8 of this does not enable us to draw any conclusion as to a similar decomposition of the set E (even when this set is an interval), unless the function considered is continuous. Two examples will now be given to show that this feature of Theorem 9.8, (which represents a restriction as compared with the results of 10, Chap. VII, is essential for the validity of the theorem. (i) Consider the function F(x)= [2 n x] / 5" , where [2 n x] denotes, as usual, n the largest integer not exceeding 2"x. This function is increasing. Its lower right- hand derivate is finite everywhere, and even, as we easily see, vanishes identically. Nevertheless, there is no decomposition of the interval J [0, 1] into a sequence of sets on which F is absolutely continuous, or even only uniformly continuous. In fact, no such decomposition can exist for a monotone function F whose points of discontinuity form a set everywhere dense in J . 294 CHAPTER IX. Derivates of functions of one or two real variables. For, if such a decomposition {E lt J 2 , ..., E n ,...} existed, one at least of the sets E n would, by Baire's Theorem (Chap. II, 9), be everywhere dense in an interval JCVo- This is plainly impossible since the function F, monotone by hypothesis, is uniformly continuous on each set E n and has points of discontinuity in the in- terior of 7. (ii) Let us now consider an example of a continuous function F(x), increasing on the interval J -^[0 9 l]^ and which has its lower right-hand derivate zero at every point of a set E, without being ACG on E. For this purpose, let us agree to call, for brevity, function attached to an interval 7 [a, 6], any function 77(#), which is continuous and non -decreasing on 7, and which fulfils the conditions: (a) . H(x) is constant on each of the intervals l k of a sequence {7^} of non- overlapping sub-intervals of 7 such that |7|=J^|7J; the length of any sub- interval of 7 on which 77(z) is constant does not exceed |7|/2; (b) H(x)U(a)-'xa and H(b)H(x)^-bx for every xtl. Such a function is easily obtained, by slightly modifying the construction of the function /(.r), considered in Chap. Ill, 13, p. 101. This being so, we shall define by induction a sequence (F n (x)\ of functions attached to the interval J OJ beginning with an arbitrary function F^(x) attached to this interval. Given the function F n attached to J , let {ZJ^faj^, b^]} be a sequence of the intervals of constancy of F n in the interval 7 . (By an interval of constancy of a function in J we mean here any interval 7C^o such that the function is constant on 7 without being constant on any sub- interval of J which contains 7 and is distinct from 7.) For each fc-=l,2,..., we determine a function 77^' (JP) attached to the interval I^f\ and we write ^ ( ' r) f77| /l) (fej /l) ) H\ n \a\ n) )} lor x*J (} 2jl*^ ft l ( f\x) --77^Vi" ) )4-4; ( - t) [77; n) (6; /l) )-Hi' l) (a; / ' ) )] for x*l k (n \ fc=l,2, ..., the sum ^-** being extended over all the values i such that b\'^ ^> JT. / The sequence (F n (x)\ being thus defined, let (9.11) The function F(x) is clearly continuous, increasing, and singular on J n . Consider the set E = U (I**), and let JT Q be any point of E. Then there n It exists a sequence {7^ |) } /I _ 1 2 of intervals each of which contains x in its interior. Plainly, for each positive integer n, Fj(b^) FJ(X Q ) = if >' -<; n, and Pj(bty~*Y x ^< b< k } n x <> if />" Hence, by (9.11), F(b ( )~F(x )^(b< x )/2" for each n, and therefore F + (x )=(). Nevertheless, the function F is not ACG on E. To see this, suppose, if pos- sible, that E is the sum of a sequence of sets E n on each of which the function F is AC. Since the set J E is the sum of a sequence of non -dense closed sets, one at least of the sets E n is everywhere dense in a sub-interval 7 of ./ , and, since the function F is absolutelv continuous on each E . this fn not inn wrmM [ 10] A theorem of Denjoy-Khintchine on approximate derivates. 295 $ 10. A theorem of Denjoy-Khintchine on approximate derivates. The considerations of the preceding will now be completed by a theorem which establishes, for the extreme ap- proximate derivates, relations similar to those which hold for Dini derivates (cf. 4). This theorem was proved independently by A. Denjoy [6, p. 209] and by A. Khintchine [4; 5, p. 212] (cf. also J. C. Burkill and U. S. Haslam- Jones [1;3]). (10.1) Theorem. If a finite function F is measurable on a set E and if, to each point x of E, there corresponds a measurable set Q(x) such that (i) the lower unilateral density of Q(x) at x is positive on at least one side of the point x and (ii) FQ( X )(X)<+OO or FQ^(X)> oo, then the function F is approximately derivable at almost all the points of E. Consequently, if a finite function F is measurable on a set E) then at almost every point of E either the function F is approximately derivable, or else Fti ) (x)==F ap (x) = + oo and Proof. In view of Lusin's Theorem (Chap. Ill, 7), we may suppose that the set E is closed and that the function F is con- tinuous on E. To fix the ideas, consider the set A of the points x E such that (i t ) the lower right-hand density of Q(x) at x is positive and (iii) FQ (X )(X)<+ oo. We shall show that the function F is ap- proximately derivable at almost all the points of A. By symmetry, this assertion will remain valid for each of the other three subsets of E, defined by a similar specification of the conditions (i) and (ii) of the theorem. Let us denote by P the set of the points of A at which the function F is not approximately derivable, and suppose, if pos- sible, that |P|>0. For each positive integer n, let A n be the set of the points x of E such that the inequality 0<A<l/n implies (10.2) \E[F(t)F(x)^n-(tx)- 9 teE\ x^t^x+h]\^hjn. t The sets A n are closed. To see this, let us keep an index n fixed for the moment, and let {#/)/=i,2,... be a sequence of points of the set A n converging to a point X Q . Let fe<l/w be a non-negative number, and, for brevity, let Ei=E[F(t) F(xt)^n-(t a?/); teE; #/<*<#/+*] t where i=0,l,2,... . We obtain \Ei\^h/n for i=l,2,..., and since, by continuity of F on J5, we have E D lim sup E h it follows (cf. Chap. I, Theorem 9.1) that |J0 |^A/n, which shows that x A n , i. e. that A n is a closed set. 296 CHAPTER IX. Derivates of functions of one or two real variables. Let us now denote, for every pair of positive integers n and fc, by A nik the set of the points x of A n such that the inequality OO<l/# implies (10.3) |E[*e An; a A<I<0]|>(1 in" 1 )-*. t We observe easily that the sets A n , and therefore the sets A n ,k also, cover the set A almost entirely. Hence, there exists a pair of positive integers n Q and fc such that |-4.,^-P|>0. Let R denote a portion of the set A^^P such that (10.4) |-R|>0, (10.5) d(R)<l/n and (10.6) d(R)<l/k . Writing G(x)=F(x)~ (n +l)-#, we shall show that the func- tion G is monotone non-increasing on R. Suppose therefore, if pos- sible, that there exist two points a and b in R, where a<b, such that (10.7) 0(a)<G(b). Let J=[a,ft], Since the set A n<) is closed and the function G continuous on A n ^ the function G attains, at a point c of the set An^Jj the lower bound of its values on this set. In virtue of (10.7) we have c<b. Since ceA^ and since, by (10.5), 0<fr c<l/n , we may put n=n , x=c and h=bc in the relation (10.2). We thus obtain (10.8) |E[0(*)0(0)< (t-c);teE;c^t^b]\^(b-c)ln . t Again, since beRCA^^ and since, by (10.6), 0<ft c<l/fc , we may put n=n , x=b and h=bc in (10.3). This gives (10.9) \K[teAn ;c^t^b]\^(l-^).(b-c). t Now the sets which occur in the relations (10:8) and (10.9) are both measurable; it therefore follows from these relations that there exist, in the open interval (c,&), points teA^ for which 0(t) (?(c)< (t c)<0. This is plainly impossible, since the function G attains its minimum on the set A^J at the point c. The function G is thus monotone on the set J?, and since it is, moreover, measurable (indeed continuous) on the closed set JS/D-B, it follows that G is approximately derivable at almost all the points of R. On the other hand, however, since JBCP the function F is approximately derivable at no point of J2, and, in view of (10.4), we arrive at a contradiction. This completes the proof. [11] Approximate partial derivates of functions of two variables. 297 By a slight modification of the proof, we may extend Theorem 10.1, in a certain way, to functions which need not be measurable. Let us agree to under- stand by approximate derivability of a finite function .F at a point a? , the existence of a set for which x is a point of outer density and with respect to which the function F is derivable (if the function F is measurable, this notion of approxi- mate derivability clearly agrees with the definition of Chap. VII, 3). When approximate derivability is interpreted thus in the statement of Theorem 10.1, this theorem remains valid without the hypothesis that the function F be meas- urable on the set E (although the hypothesis concerning the measurability of the sets Q(x) remains essential). From Theorem 10.1, we may deduce the following proposition: //, for a finite function F, we can make correspond to each point x of a set E, a measurable set Q(x) whose lower right-hand density at x is positive, and with respect to which the junction has an infinite derivative at x, then the set E is of measure zero. This theorem is similar to Theorem 4.4, but only partially generalizes the latter. It is not actually possible to replace, in Theorem 4.4, the ordinary, by the approx- imate, limit, without also removing the modulus sign in the expression \F(x+ h)~F(x)\. This rather unexpected fact was brought to light by V. Jarnlk [2], who showed that there exist continuous functions F for which the relation lim ap \F(x+h) F(x)\/h=+oo holds at almost all points x. Finally, let us note that Theorem 10.1 is frequently stated in the following form: If a finite function F is measurable on a set E, then at almost every point x of E either (i) the function F is approximately derivable, or else (ii) there exists a meas- urable set E(x) whose right-hand and left-hand upper densities are both equal to 1 at x, and with respect to which the two upper unilateral derivates of F at x are -{-co and the two lower derivates oo. It has been shown by A. Khintchine [4] (cf. also V. Jarnlk [2]) that there exist continuous functions for which the case (ii) holds at almost every point x. 11. Approximate partial derivates of functions of two variables. The which follow will be devoted to generali- zations of the results of 4 for functions of two real variables (their extension to any number of variables presents, as already said, no fresh difficulty). In this we shall establish some subsidiary results. Given a plane set Q and a number 77, we shall understand by the outer linear measure of Q on the line y=r/, the measure of the linear set E[(J, r?) cQ]. Similarly, we define the outer linear measure t of Q on a line #=f, where f is any number. It follows from Fubini's Theorem in the form (8.6), Chap. Ill, that if Q is a measurable set whose linear measure on almost all the lines yri (i.e. on the lines y=rj for almost all values of rj) is zero, then the set Q is of plane measure zero. 298 CHAPTER IX. Derivates of functions of one or two real variables. A point (xQ,y Q ) w iM ^ e termed point of linear density of a plane set Q in the direction of the x-axis, if x is a point of density of the linear set E[(J,y )f $]. We define similarly the points of linear den- t sity of Q in the direction of the y-axis. (11.1) Theorem. Almost all points of any measurable plane set Q are points of linear density for it both in the direction of the x-axis and in that of the y-axis. Proof. We may clearly assume that the set Q is closed. Con- sider, to fix the ideas, the set D of the points of Q which are points of density of Q in the direction of the #-axis. Since the set Q D is of linear measure zero on each line y= r\, the proof of the relation \Q D\ = Q reduces to showing that the set D is measurable. In order to do this, we write, for each point (x,y) and each pair of numbers a and fc, and we denote, for each pair of positive integers n and &, by Q,,^ the set of the points (x,y) of Q such that the inequalities a<x<b and ba^ljk imply \E(x,y\ a,b)\^(l n~ l )-(b a). Plainly D=HSQn t k. n k We now remark that all the sets Q n>k are closed. To see this, we keep the indices n and k fixed for the moment, and consider an arbitrary sequence {(Jo l9 y l )} i=l 2 f points of Q n ^ which converges to a point (x ,y Q ). Let a and b denote real numbers such that a<x <b and ba^l/k. For every sufficiently large index i, we then have a<x;<b, and so \E(x i9 y f m 9 a,fe)|>(l n~ l )-(b a). Now it is easy to see that lim sup E(x p t/.; a,6)Cl?(# ,t/ ; a,6); it therefore follows from Theorem 9.1, Chap. I, that |jE(# ,# ; #,fc)|^(l n l )-(ba), and so, that (# , 3/ ) Q n ,k> Since the sets Q n ,k are closed, D is a set ($<,<>) and this com- pletes the proof. If F is a finite function of two variables, the extreme approx- imate partial derivates of F(x,y) with respect to x will be denoted by F*p x , F^ x , Fap x and jPapj. If these derivates are equal at a point (x,y)j their common value, i.e. the approximate partial derivative of F with respect to a?, will be denoted by F'^ x (x,y)- Analogous symbols will be used with respect to y. For the partial Dini deri- vates, we shall retain the notation of Chap. V, namely F*, F, etc. [11] Approximate partial derivates of functions of two variables. 299 (11.2) Theorem. If a finite function of two variables F is meas- urable on a set Q, its extreme approximate partial derivates are them- selves measurable on Q. Proof. In view of Lusin's Theorem (Chap. Ill, 7), we may suppose that the set Q is closed and that the function F is con- tinuous on Q. Consider, to fix the ideas, the derivate F^ x . Let a be any finite number and let P be the set of the points (x,y) of Q at which Ffp x (x,y) ^ %. We have to prove that the set P is measurable. For this purpose, let D denote the set of the points of the set Q which are its points of linear density in the direction of the ff-axis. Further, for every point (x,y) and every positive integer n, let E n (x,y) denote the set of the points t such that *>*, (t,y)cQ and F(t,y) -F(x,y) < (a +n~ l ) (t-x). We easily observe (cf. Chap. VII, 3) that, in order that ^apj(#o>?/o) ^ fl a t a point (x ,y Q ) e Z), it is necessary and suffi- cient that the point (x^y^) be a point of right-hand density for every set E n (x^y^ where n = 1,2,... Hence, denoting for every system of three positive integers n, k and p, by Q n ,^ P the vset of the points (x,y) of Q such that the inequality O^A < 1/p implies \En(*,y)-[x,x+h]\^(l-k~*)-h, we have n k p Now the set Q is closed and the function F is continuous on Q, and by means of Theorem 9.1, Chap. F (cf. the proofs of Theorems 10.1 and 11.1) we easily prove that all the sets Q n ,k, P are closed. Hence, by (11.3), the set P-D is measurable, and since, by Theorem 11.1, \Q Z)| = 0, we see that the set P is measurable also. This completes the proof. It follows, iii particular, from Theorem 11.2 that the extreme approximate derivates of any finite measurable function of one real variable are themselves mea- surable functions. We thus obtain a result analogous to Theorem 4.3, Chap. IV, which concerned the measurability of Dini derivates (cf. also Theorem 4.1, Chap. V. and the remark p. 171). 300 CHAPTER IX. Derivates of functions of one or two real variables. $ 12. Total and approximate differentials. A finite func- tion of two real variables F is termed totally differentiabk, or simply differentiable, at a point (# ,# ) if there exist two finite numbers A and B such that the ratio (12.1) [F(x J y)^F(x Q ,y Q )^A>(x^x^B^(y^y.)]![\x^x \ + \y^y Q \] tends to zero as (, y )->(#< #o)- ^^ e P a * r * num l> ers {,} is then termed total differential of the function F at the point (x^y^) and we see at once that .4 and B are the partial derivatives of F at (j? ,// ) with respect to x and to y respectively. If, for a finite function of two variables F and for a point (a?o#o)> there exist two finite numbers A and B such that the ratio (12.1) tends approximately to as (#,y)->(# jy<))> the function F is termed approximately differentiate at (x Q ,y Q ) and the pair of numbers {A,B} is called approximate differential of F at (# ?2/o)- The numbers A and B will be called coefficients of this differential. We see at once that no function can have at a given point ^ than one differential, whether total or approximate. The existence of a total differential of a function F(x,y) at a point may be interpreted as the existence of a plane, tangent at this point to the surface zF(x,y) and non -perpendicular to the xy-plane. In this way the notion of total differentiability of functions of two variables corresponds exactly to the similar notion of derivability of functions of one variable. Nevertheless, whereas every func- tion of bounded variation of one variable is almost everywhere derivable, a function of bounded variation (in the Tonelli sense), and even an absolutely continuous function, of two variables may be nowhere totally differentiate (cf. W. Stepanoff [3, p. 515]). The coefficients of an approximate differential of a function at a point are not, in general, approximate partial derivatives of this function. Nevertheless they coincide with the latter almost everywhere, as results from the following theorem: (12.2) Theorem. In order that a finite junction of two variables JP, which is measurable on a set Q, be approximately differentiate at almost all the points of this set, it is necessary and sufficient that F be, almost everywhere in Q, approximately derivable with respect to each variable. When this is the case, the approximate partial derivates F f &Px (x,y) and Ja Py (#,y) we, at almost all the points (x.y) of Q, the coefficients of the approximate differential of F* [ 12] Total and approximate differentials. 301 Proof. 1 Suppose that the function F is approximately differentiate at almost all the points of Q. We denote, tor each positive integer n, by R n the set of the points (f, ??) of Q such that, for every square J containing (, 77), we have (12.3) whenever <5(<7)<2/n. Writing R=%R n , we clearly have \Q R\ = Q. n Let us now denote, for a general plane set F and any number 17, by E [r}] the linear set of the points f such that (^^tE. Keeping fixed, for the moment, a positive integer n and a real number *? , we consider any two points 3 and 2 of R [ ^ for. which 0^ 2 fi<l/w , and we denote by J the square [!i, 2 ; 7 /o? r /o+^2~ fil- We then have (5(J )^:2/n , and so, putting n = w , J=J& ^=^ i n (12.3). and choosing f= f x and f=f z successively, we see at once that the square J contains points (x,y) for which we have at the same time and Hence |^(f 2 >*7o) ^(fi^o)! < 4n o * ta fil> which shows that, for any fixed 77, F(x,r]), as a function of #, is AC on each set /?L >;I > and so VBG on the whole set R M (cf. Chap. VII, 5). Now R is (with Q) a plane measurable set, so that the linear set R 1 ' 1 * is meas- urable foi almost every ??. Hence (cf. Theorem 4,3, Chap. VII) for almost all 77, the function F(x,t]) is approximately derivable with respect to x at almost all the points of R [n} . Since further, by Theorem 11.2, the set of the points of R at which the function J^ is approximately derivable with respect to one variable, is meas- urable, it follows at once that the function F is approximately derivable with respect to x at almost all the points of R, and so, at the same tiipe, at almost all the points of Q. Similarly, we establish the corresponding result concerning approximate derivability of F with respect to y. 2 Suppose that the function F is approximately derivable, at almost all the points of Q, with respect to x and with respect to y. We shall show that the function J? then has, at almost all the points of Q, an approximate differential with coefficients F't Px (x,y) and F^ y (x,y). On account of Theorem 11.2 and of Lusin's Theorem (Chap. Ill, 7), we may suppose that 302 CHAPTER IX. Derivates of functions of one or two real variables. (a) the set Q is bounded and closed, (b) the function F is approximate- ly derivable with respect to each variable at all the points of Q, and (c) the function F, and both its approximate partial derivatives, are continuous on Q. This being so, we write, for each point (,17) of Q and each point (x,y) of the plane, (12.4) A(f , r, *) = \f(a>, i)-f(S, i?)- (* - D,(f , 17; y) = c and T be any positive numbers. We shall begin by de- fining a positive number a and a closed subset A of Q such that |$ A\<e and such that, for any point (,?), whenever (, ??)e^i, a^f^fe and ft a<o. For this purpose, let us denote, for each positive integer n, by ^t n the set of the points (f, ??) of $ such that the inequality in the first line of (i) is fulfilled whenever a<<b and ft <l/n. Since the set Q is closed and since the function F and its derivatives F' a ^ x and .Fap are continuous on Q, it is easily seen that all the sets A n are closed. On the other hand, the sets A n form an ascending se- quence and we immediately see that the set Q lim A n is of n measure zero on each line yrj. Hence, this set being measurable, we have \Q lim .4^ 0. Consequently \Q A no \<e for a suffi- n ciently large index n , and writing a=lln and A = A no we find that the inequality \Q A\<e and the condition (i) are both satisfied. In exactly the same way, but replacing the set Q by A and interchanging the role of the coordinates x and y, we determine now a positive number a } <a and a closed subset B of A such that \AB\<e and that for any point (,77) whenever (,77)6,8, a<?7^ft and b a<a r Finally, let a 2 <a 1 be a positive number such that for any pair of points (x l9 yj and (# 2 , y 2 ) of Q subject to the con- ditions \x 2 a?,|<<y t and |y t [ 12] Total and approximate differentials. 303 This being so, let (| , ?? ) be any point of B. Let J \oufa a 2 ,/y denote any interval such that ( >*?o)^ and 6(J)<a 2 <a 1 <a. We write: .v and. for each Then any point (#,y) such that yeE 2 and xeE^y) belongs 'to the set Q-J and, for such a point, we have On the other hand, it follows at once from (ii) and (i) respectively, that |E 8 |>(1 fi)-(/S a -a 8 ), and |\(y)|>(l -)(& -a,) whenever ye #2- Hence, J5(| , >/ ; ,r,//) being a measurable (indeed contin- uous) function of the point (j?,#) on (?-J, it follows that the set of the points (x,y)tQ-J such that D(f ,?7 ; ff,yK2T-|> f |+!# ~ '/oil is of measure at least equal to (1 f) 2 (/tf, a t ) ( 2 a 2 )= (1 f) 2 -|J|. The point ( , ?/ ) here denotes any point of the set JB, and e/ any interval, containing ( , /y ), whose diameter is sufficiently small. Therefore, since K? Z?|^|y A\+ \A ~B\^.2ej where e is at our disposal, we see that, for every positive number r, almost every point (f, YI) of Q is a point of density for the set of the points (x<y) of Q which fulfil the inequality />(, /;; jr,y)l[\f \ + \yrj\]^2rj and in view of (1^.4), this completes the proof. We notice a similarity between the preeedintf proof and that of the "Den- sity Theorem*' (Chap. IV, 10). Actually the result just established constitutes a direct generalization of the Density Theorem. To see this, we need only inter- pret, in the statement of Theorem 12.2, the function F as the characteristic function of the set Q (cf. the first edition of this book, p. 231). The notion of approximate differential, together with Theorem 12.2, are due to W. Stepanoff (3). There is, however, a slight difference between the defi- nition adopted here and that of Stepanoff, so that, in its original form, as proved by Stepanoff, Theorem 12.2 generalizes Theorem 6.1, Chap. IV, rather than the Density Theorem of 10, Chap. IV. We conclude this by mentioning the following theorem, which, in view of Theorems 9.0 and 11.2, is an immediate con- sequence of Theorem 12.2: 304 CHAPTER IX. Derivates of functions of one or two real variables. (12.5) Theorem. Suppose that a finite function of two variables F which is measurable on a set Q, has at each point of Q at least one finite Dini derivate' with respect to x and at least one finite Dini derivate with respect to y. Then the function F is approximately differentiate at almost every point of Q. 13. Fundamental theorems on the contingent of a set in space. Following F. Roger [2], we shall now extend to Bets in the space jR 3 , the results obtained in 3. The proofs will be largely a repetition of those of 3 with the obvious verbal changes. We shall therefore present them in a slightly more condensed form. Generalizing the definitions of 3, p. 264, to functions of two variables, we shall say that a function F(x,y) fulfils the Lipschitz condition on a plane set E, if there exists a finite constant N such that \F(x 2J t/ 2 ) F(x yj\ < N- [\x 2 x l + \y. l yj] for every two points (#1,^1) and (x 2J y 2 ) of E. We verify at once that the graph of the function F on E is then of finite area whenever \E\<+oo, and of area zero whenever, in particular, |JE|=0 (cf. Chap. II, 8; more precisely, we have, for every set E, In the sequel we shall make use of the following notation for limits relative to a set. If E is a set (in any space) and t Q is a point of accumulation for I?, the lower and upper limits of a function F(t) as t tends to t Q on E will be written liminf^JF 7 ^) and limsup^^) *-*** /-H, respectively. Their common value, when they are equal, will be written ]imsF(t). t-+t (13.1) Lemma. Let R be a set in the space JB 3 , a fixed direction in this space and P the set of the points a of R at which contg/a con- tains no half-line of direction 6. Then (i) the set P is the sum of a sequence of sets of finite area and (ii) at each point a of P, except at most at those of a subset of area zero, the set R lias an extreme tan- gent plane, for which the side containing the half -line ad is its empty side. In the particular case in which 6 is the direction of the positive semi-axis of z, the set P is expressible as the sum of an enumerable infinity of sets each of which is the graph of a function on a plane set on which the function fulfils the Lipschitz condition. [$13] Fundamental theorems on the contingent of a set in space. 305 Proof. We may clearly suppose (in the first part of the theo- rem also) that is the direction of the positive semi-axis of z. We denote, for every positive integer n, by P n the set of the points (x,y,z) of P such that the inequalities \x' a?|<l/w, |y' y|<l/n and |' s|<l/n imply z' <-[|o?' o?| + |y' y|] for every point (#',y',z') of #. We express, further, each P n as the sum of a sequence {P n ,*}*=i,2,... of sets with diameters less than 1/n. For every pair of points (x^y^z^ and (# 2 ,j/ 2 ,2 2 ) of the same set P /I>/r , we thus have |t il<w-[|o; a o?,| + |y a yj], and if we denote by Q n .k the orthogonal projection of P,,,/, on the #y-plane, we easily see that the set P nt k may be regarde'd as the graph of a function f\ ltk on Q /t , fc . Plainly \F nf k(^y 2 )Fn,k(^y l )\<n'[\Jc 2 ~x l \ + \y z ---y l \] for every two points (#!,2/i) and (r a ,;y a ) of ^ ;/ ^. Thus F n ^ fulfils the Lipschitz con- dition on Q n , k and hence (of. p. 304) ^ a (Pn.*)=^2{B(-P.*;<?n.'*))<+ 00 - Thus P=P n ,k is the required expression of the set P. /i,A It remains to discuss the existence of an extreme tangent plane to K at the points of P. For a fixed pair of positive integers w and k, the function F n ,^ which fulfils the Lipschitz condition on the set #,/,, can be continued at once, by continuity, on to the closure <?,* of this set, and then on to the whole plane by writing F n k (x,y)=Q outside Q, ltk . On account of Theorem 1U.2, the function F n ,k is approximately differentiate at almost all the points of $ /.. Hence, denoting by Q, itk the subset of Q,,,tt consisting of the point* of density of Q n .k at which F n ^ is approximately ilifferentiahl(, we see that \Q,,,k-Qn t k\ = Q and hence, that Az{tt(F .*&.* --$.*)!- -0. We need, therefore, only show that K has an extreme tangent plane at each point of B(F,,,*; (?,*), and that, further, the half-line with the direction of the positive semi-axis of z is contained in the empty side of this plane. Let ( %o) he ari y P oillt ot ' B(/V *;.*) <! lrt {-'o'Ai) be the approximate differential of F n ,k at the point ( , //). Let f<l be any positive number, and let # f be the set of the points (J?ftn.* such that Since the function F n , k is measurable, (f u , /; ) is (cf. ( 1 ha]>. VII, , r >) a point of outer density for the net /,. Hence we can make cor- respond to each point (,>/, C), sufficiently near to ( ''/o-^o) a such that: 306 CHAPTER IX. Derivates of functions of one or two real variables. (13.2) If-fJ^f-fJ and to' (13.3) |*'-|<H*-ol and h' Bemembering that Co=^'n,*(fo J ?o)> we now write for brevity V a ',*)')=*'n,k(S',ri')-t-A .(('-g )-B -(r i '-v). We thus have This being so, let (, rj, C) be a point of R such that each of the differences |f f c |, |/ j/J and |C C | is less than, or equal to, l/4 2 . Then by (13.3), we have |' f|<l/n and \r)' y|<l/n, while, by (13.2), |J T n .*(l',7 ? ')-fol<-CII'-fol+!'?'--'?ol]<l/2n ! and so !*.*(', if') C|<l/n. Since the point (|',T?', _F n , A (|',j?')) belongs to B(^;,)CP n , it follows that f *.({', ?')< [|f-*'|+b-7'|], and, again making use of (13.3), we deduce from (13.4) that (13.5) We now observe that, since (', V) 6 ^> (13.2) implies Hence, being an arbitrary positive number, we derive from (13.5) (13.6) lim Moreover, since {^I ,fi o y is the approximate differential of the func- tion F n ,k at (f ^o) an( l since the point ( *?o) i H a point of outer density for the set Q ntk , the plane z ^A^(x f ) jB -(i/ ^ ) = is certainly an intermediate tangent plane (cf. 2, p. 263) of ^? at the point (f u >*?o>o)- I* w therefore, by (13.6), an extreme tangent plane at thifi point, with an empty side consisting of the half-space z f >^o'( ;r ~o)+^V(# *7o)- This completes the proof. We shall employ in space a terminology similar to that of the plane (cf. 3, p. 264) and agree to say that the contingent of a set KCR^ at a point a of K is the whole apace if it includes all the half-lines issuing from the point a; and again, that the contingent of E at a point a of E is a half-*pace, if E has an extreme tangent plane at a and if contg^a consists of all the half-lines issuing from a which are situated on one side of this plane. We make use of these terms to state the analogue of Theorem 3.6: [| 13] Fundamental theorems on the contingent of a set in space. 307 (13.7) Theorem. Given a set R in /J 3 , let P be a subset of R at no point of which the contingent of R is the whole space. Then (i) the set P is the sum of an enumerable infinity of sets of finite area and (ii) at every point of P, except at those of a set of area zero, either the set R has a unique tangent plane, or else the contingent of R is a half -space. The proof of this statement, which follows directly from Lemma 13.1, is quite similar to that of Theorem 3.6. We need only replace, in the proof of the latter, the terms length, tangent and half -plane by area, tangent plane and half -space, respectively. It only remains to extend to space, Theorem 3.7. This ex- tension, in the form (13.11) in which we shall establish it, is essen- tially little more than an immediate, and almost trivial, conse- quence of Theorem 3.7. Its proof requires however some subsidiary considerations of the measurability of certain sets. (13.8) Lemma. If Q is a set (ft^) ^ n ^3* ^ s orthogonal projection on the xy-plane is a measurable set. Proof. Let us 'denote generally, for every set E situated in J? 3 , by I\E) its projection on the #y-plane. In order to establish the measurability of the set F(Q), it will suffice to show that for each >0 there exists a closed set PCT(Q) such that \P\^\F(Q)\- B. We express Q as the product of a sequence {Q n }n-i.2. of sets (5 a ). It may clearly be assumed that the set Q is bounded and that, moreover, all the sets Q n are situated in a fixed closed sphere # . We shall define in jR 3 , by induction, a sequence {-P M } n -o,i,.... of closed sets subject to the following conditions for n=l,2,...: (i) FnCF^, (ii) F n CQn and (iii) \r(F n -Q)\^ F(F n ^Q)\-el2 H . For this purpose, we choose F =8 and we suppose that the next r 1 sets F n have been defined. We have QCQ n and so F r -i-<i,'<i=F r -rQ, and since F r -rQ r i*, with Q r , a set (ft,), there exists a closed set F,CF r -rQr such that \l\F t -Q)\^\r(F r r 0)i~e/2 r . This closed set F r clearly fulfils (i), (ii) and (iii) for n = r. Now let F=nF n = \imF n . It follows from (ii) that FCQ, and therefore that r(F)cr(Q). Further, F(F) is a closed set, for, since {F n \ is a descending sequence of closed and bounded sets, we easily see that r(F)=limr(F n ). Finally this last relation coupled with (iii) shows that \r(F)\^lim\r(F tt 'Q)\>\r(F Q 'Q)\-f^\r(Q)\-e, which com- n pletes the proof. 308 CHAPTER IX. Derivates of functions of one or two real variables. It would be easy to prove that the projection of a set (5 a< j) is the nucleus of a determining system formed of closed sets and thus to deduce Lemma 13.8 from Theorem 5.5, Chap. II. We have preferred, however, to give a direct ele- mentary proof, based on a method due to N. Lusin [3]. The same argument shows that any continuous image of a set (5^) * 8 measurable. It has been proved more generally (vide, for instance, W. Sierpinski [II, p. 149], or F. Hausdorff [II, p. 212]) that any continuous image of an analytic set -(in particular, of a set measurable ($})) situated in 7f 8 is an analytic and, therefore, measurable set. (13.9) Lemma. Given a set R in /2 3 , let Q be the set of the points (f, TI, C) of R which fulfil the condition: (A) the part of the contingent of R at the point (, 77, t), which is situated in the plane # , is wholly contained in one or other of the two half -spaces y^rj and y^rj* Then the orthogonal projection of the set Q on the xy-plane is of plane measure zero. Proof. We may clearly suppose that the set R is closed (for the contingent of any set R coincides, at all points of R, with that of the closure of R). Let us denote generally, for any set E in 12 3 and any number f , by E m the set E[(,y,z)cE]. It follows from Theorem 3.6 that, <.v,*) for every f , the plane set R [ * ] has an extreme tangent, parallel to the z-axis at every point of Q { ' ] except those of a set of length zero. Hence, by Theorem 3.7, the projection of Q on the #t/-plane is of linear measure zero on each line x f of this plane, and, in order to prove that this projection is of plane measure zero, we need only show that the latter is measurable. Let us denote, for each pair of positive integers k and n, by A^n the set of the points (f, ??, C) of R such that the inequalities (13.10) \x-S\ + \y-r t \ + |*-C|<l/n and imply, for any point (x,y,z)eR, the inequality y r Similarly, we shall denote by #/,, the set of the points (f, 17, ) of R for which the inequalities (13.10) imply, for every point (x, y, z) of R, the inequality y<q^-.[\x\ + \zl;\]llc> Writing A=HZA k , n and # ft n we find that Q^=A + B. On the other hand, since tne set R is, by hypothesis, closed, we observe at once that each set A k , n , and like- wise each set #*,, is closed. The sets A and B, and so the set ^ also, are thus sets (5 at >)> and in view of Lemma 13.8, the projection of Q on the #i/-p!ane is a measurable set. f14] Extreme differentials. 309 (13.11) Theorem. Given a set R in JR 3J let P be a subset of R at every point of which the set R has an extreme tangent plane parallel to a fixed straight line D. Then the orthogonal projection of P on the plane per- pendicular to D is of plane measure zero. Proof. We may clearly suppose that the straight line D is the z-axis. Let us denote by P l the set of the points of P at which the extreme tangent plane, parallel, by hypothesis, to the z-axis is not, however, parallel to the yz-plane. Similarly, P 2 will denote the set of the points of P at which the extreme tangent plane is not parallel to the xz-plane. We then have P=P 1 + P 2 . Now we observe at once that each point (, ?/,C) of P l fulfils the condition (A) of Lemma 13.9. It therefore follows from this lemma, that the projection of P l on the #z/-plane is of plane measure zero. By symmetry, the same. is true of the projection of the set P 2 . The proof is thus complete. $ 14. Extreme differentials. Let F be a finite function of two real variables. A pair of finite numbers {A,B} will be called upper differential of F at a point (x oj y ) if, when we write =jP (# ,y ), (i) the plane z Z Q A-(x x Q ) + B-(y y Q ) is an intermediate tan- gent plane of the graph of the function F at the point (# ,y ,2 ) and (ii) lim sup ?&*)-*(* yo--B-Cjf-lfo) These conditions may clearly be replaced by the following: (i x ) the plane z z =A'(x X ) + B-(y y ) is an extreme tangent plane of the graph of F at (# ,y ,z ) with the empty side z z^A-(x x Q ) + B-(y y ), and (ii^ lim sup F(x J y)^F(x QJ y Q ). (ar,0)-Mjr,y ) The definition of lower differential is similar, and the two differentials, upper and lower, will be called extreme differentials. If a function F has a total differential (cf . 12, p. 300) at a point, this differential is both an upper and a lower differential of F at the point considered. Conversely, if a function F has at a point (#o>yo) both an upper and a lower differential, these are identical and then reduce to a total differential of F at (x OJ y Q ). For a finite function of one real variable F, the existence of an upper dif- ferential at a point x is_to be interpreted to mean that F + (x Q )~F~(x )4 oo (in which case the number F*(x<>)=F~(x Q ) may be regarded as the upper differential of F at x ). There is a similar interpretation for the lower differential of functions of one variable. This interpretation brings to light the relationship between tho theorems of this and those of 4. 310 CHAPTER IX. Derivates of functions of one or two real variables. We propose to give an account of researches concerning the existence almost everywhere of total, approximate, or extreme differentials. These re- searches were begun by H. Radeniacher [3], who established the first general sufficient condition in order that a continuous function be almost every- where differentiate. W. Stepanoff fl;3] later removed from Rademacher's reasoning certain superfluous hypotheses, and obtained a more complete result, valid for any measurable function: In order that a junction F which is meas- urable on a set K, should be differentiate almost everywhere in E, it is necessary and sufficient that the relation lim sup \F(r,y) F(,i?)\/[\x f|-f \y ?;|]<-|-oo should hold at almost all the points (,*?) of E. (Certain details of Stepanoffs proof, par- ticularly those concerning measurability of the Dini partial derivates, have been subjected to criticism (cf. J. C. Bur kill and U. S. II aslam -Jones [1].) U. S. II a slam -Jones [1] extended further the result of Stepanoff, and by introducing the notion of extreme differentials (which he called upper and lower derivate planes), obtained theorems analogous to those of Denjoy for functions of one variable. The researches of Haslarn- Jones have been continued and completed by A. J. Ward [ 1 ; 4] who, in particular, removed the hypothesis of measur- ability in certain of Haslam-Joues's theorems. We shall derive the results of H aslam -Jones from the theorems of the preceding (cf. F. Roger [3]; direct proofs will be found in the memoirs of H aslam -Jones and W r ard referred to, and in the first edition of this book). In what follows, we shall make use of some subsidiary con- ventions of notation. If F is a function of two real variables and t denotes a point (x,y) of the plane / 2 , we shall frequently write F(t) for F(x,y). If t l = (x lj y l ) and t 2 ~(x 2 ,y 2 ) are two points of the plane, \t 2 j will denote the number \x 2 x t \ + \y z 3/il- Given in the plane two distinct half -lines issuing from a point < , each of the two closed regions into which these half-lines divide the plane will be called angle. The point t will be termed vertex of each of these angles. We shall begin by proving a theorem somewhat analogous to Theorem 1.1 (ii). (14.1) Theorem. Let F be a finite function in the plane H 2 and let E be a plane set, each point T of which is the vertex of an angle A(r) such that lim&up4(,)f r '(0<Um sup<F(t)- Then the set E is of plane t->r /->r measure zero. Proof. Let us denote, for each pair of integers p and q, by E p ^ the set of the points T of E at which ttm&\ip A(r )F(t)<plq<]im8\ipF(t). t-+r t~+r For fixed p and </, we observe that no point reEp^ is a point of accumulation for the part of the set E p , q contained in the interior of the corresponding angle A(r). Hence, no point of the set E p%q can be a point of outer density for this set. Each of the sets E p , q is thus of plane measure zero, and the same is therefore true of the whole set E. [14] Extreme differentials. 311 As we easily see, in virtue of Theorem 3.6, each set E p , qt and consequently the whole set E, is the sum of a sequence of sets of finite length (this of course, implies that E is of plane measure zero). Cf. A. Koimogoroff and J. Ver6enko [I]. (14.2) Theorem. Let F be a finite function in the plane. Then (i) if P is a plane set each point r of which is the vertex of an angle A(r) such that (14.3) Hm A(t) \F(t)-F(T)\/\t-T\=+oo, f-K the set P is necessarily of plane measure zero; (ii) if Q is a plane set each point r of which is the vertex of an angle A O (T) such that (14.4) lim wp An(f) [F(t)-F(T)]l\t-T\<+oo, /-T the function F necessarily has an upper differential at almost all the points of Qj (iii) if R is a plane set each point T of which is the vertex of two angles A^r) and A%(T) such that lim su P/MT) [F(t)-F(r)]l\t~r\<+ oo and lim infx, ( r the function F is totally differentiable at almost all the points of R. Proof, re (i). By Theorem 13.7 the set B(^;P) has, at each of its points except those of a subset of area zero, an extreme tan- gent plane. The latter is seen to be necessarily parallel to the c-axis. Hence, by Theorem 13.11, the set P, as the projection of J$(F;P) on the ;n/-plane, is of plane measure zero. re (ii). It clearly follows from (14.4) that, at each point r of Q, we have limsup^ )( r)^(0<^( T )- Hence, by Theorem 14.1, we have /->t lim sup F(t)^.F(r) at all the points r of {?, except at most those /->r of a set Q Q of measure zero. Let us now denote by & the graph of the function F (on the whole plane). Let R^ be the set of the points of B(F,Q) at which the set B has no extreme tangent plane, and B 2 the set of the points of B(F;Q) at which such a tangent plane exists, but is parallel to the z-axis. Finally, let Q l and Q 2 l>e the projections of the sets #, and # 2 respectively, on the #t/-plane. On account of Theorem 13.7, 312 CHAPTER IX. Derivates of functions of one or two real variables. we easily verify that y! 2 (J8 1 ) = 0, and so, that \Qi\ = Q. Similarly. it follows at once from Theorem 13.11 that |Q 2 | = 0. Now, if (f, rj) is any point of Q (Q) + Q 2 ), the set B has at (f, ^^(f, y)) an ex- treme tangent plane of the form z f = Jf (f , ??)(# ) + .#(, rj)-(y 17), where Jf(, 77) and N(,rj) are finite numbers. We observe further without difficulty that the half-space is an empty side of this plane. Hence (cf. p. 309), at each point (,>?) of the set Q-(Q Q +Q, + Q 2 ), the pair of numbers {3f( ,/?),#( ,*?)} is an upper differential of the function F. This completes the proof, since | Finally, (iii) is an immediate consequence of (ii). In the case in which the function F is measurable, we can complete part (i) of Theorem 14.2 (which itself generalizes Theorem 4.4). Thus, if F is any measurable function of two variables, the set of the points (x,y) at which lim \F(x+ h,y) F(x t y)\/h+ oo, is of plane measure zero. This proposition plainly follows from Theorem 4.4, except for meas- urability considerations, essential to the proof, which seem to require general theorems on the measurability of the projections of sets (*) (cf. p. 308). We conclude with the following theorem (ct. A. J. Ward [1] and the first edition of this book, p. 234) which, in view of Theo- rem 14.2 (i), (ii), may be regarded as an extension of Theorem 9.9 to the functions of two variables: (14.5) Theorem. If F is a finite function of two variables, which is measurable on a set E and which has an extreme differential at each point of a set QC E, then this differential is, at the same time, an approximate differential of F at almost all the points of Q. Proof. On account of Lusin's theorem (Chap. Ill, 7) we may clearly suppose that the set E is closed and that the function F is continuous on E. Let us suppose further, for definiteness, that the function F has an upper differential at each point of Q, and let us denote, for each positive integer n, by Q n the set of the points / of Q such that, for every point t', \t't\ <l/w implies the inequality F(t' )- F(t)<n-\t' 1\. Finally, let each set Q n be ex- pressed as the sum of a sequence {<?,*}*_- 1,2, .. of sets with diameters less than 1/ft. We shall have Q n,k [U] Extreme differentials. 313 We see at once that the function F fulfils the Lipschitz con- dition on each set <?,*, and therefore also on each set Q n , k . Hence, by Theorem 12.2, the function F has the approximate differential \Fw x (>y)j J^ap y (x,y)} at almost every point (x,y) of each set $,*, and therefore at almost every point (x,y) of the set Q. Let us, on the other hand, denote, for each point (x,y) of Q, by {A(x,y)< B(x,y)} the upper differential of F at this point. It follows at once from the definition of upper differential, p. 309, that Fx(^y)>A(x J y)^F^(x J y) J and similarly F~(x,y)^B(x,y)^Fy(x,y), at each point (x,y) of Q. Hence, at each point (x } y) of Q at which the approximate partial derivates F^ x (x,y) and F* py (x,y) exist, we have -l(o?,y)=Ja Pjc (0,y) and B(x 1 y)=F^ if (x J y). The upper differential (A(x,y), B (x^y)} of the function F thus coincides at almost all points (x,y) of Q with the approximate differential of F. NOTE I. On Haar's measure by Stefan Banach. 1. This Note is devoted to the theory of measure due to Al- fred Haar [1]. Haar's beautiful and important theory deals with measure in those locally compact separable spaces for which the notion of congruent sets is defined. His measure fulfils the usual conditions of ordinary Lebesgue measure: congruent sets are of equal measure and all Borel sets (more generally, all analytic sets) are measurable. The theory has important applications in that of continuous groups. To complete the definitions of Chap. II, 2, we shall say that a set situated in a metrical space is compact, if every infinite subset of the set in question has at least one point of accumulation. A met- rical space is termed locally compact if each point of this space has a neighbourhood which is compact. 2. In what follows we shall denote by E a fixed metrical space, separable and locally compact, and we shall suppose that, for the sets situated in Z, the notion of congruence s is defined so as to fulfil the following conditions: 1 1 . A = B implies BzzA; A^B and B^C imply J. C; 1 2 . If A is a compact open set and A?=B, then the set B is itself open and compact; 1 3 . If A^B and {A n } is a (finite or infinite) sequence of open compact sets such that AC^A^ then there exists a sequence of sets n {B n ) such thai BCZjB n and such that A n ^B n for n = l,2,...; n 1 4 . Whatever be the compact open set A, the class of the sets con- gruent to A covers the whole space JB; 1 5 . // {S n } is a sequence of compact concentric spheres with radii tending to 0, and {(?} is a sequence of sets such that Q n ^S n , then the relations a=lima /J and 6 = lim& /1 , where <*<? and &/,<?, imply a=6. On Haar's measure. 315 Given two compact open sets A and B, the class of the sets congruent to A covers, by J 4 , the set B. It therefore follows from the theorem of Borel-Lebesgue that there exists a finite system of sets congruent to A which covers B. Let h(B,A) denote the least number of sets which constitute such a system. It is easy to show by means of i l i s that, for any three com- pact open sets A, B and C, the following propositions are valid: iip COB implies ii 3 . B = C implies ii 5 . If Q(A,B)>Q and if {S n } is a sequence of compact con- centric spheres with radii tending to 0, then there exists a positive integer N such that, for every n>N, (3.1) All these propositions are obvious, except perhaps ii 5 . To prove the latter, let us suppose, if possible, that there exists an increasing sequence of positive integers {n,} such that (3.1) does not hold for any of the values n=n f . There would then exist a se- quence of sets {Gi} such that G^^ 8 n while A-G/=t=0 and 5-<?/=t=0. Consider now arbitrary points aieA-Gj and fr/e B-G ( . Since the sets A and B are compact, the sequences {a/} and {b t } contain re- spectively convergent subsequences {a i } and {b^}. Let a = limav j J j j and & = lim&v. By i 5 we must have a=fe, and this is impossible j J since, by hypothesis, Q(A,B) = Q. We shall now suppose given a fixed compact open set G and a sequence {S n } of concentric spheres, with radii tending to 0, which are situated in G and therefore clearly compact. For every compact open set A, we write We then have, by ii 4 , n ) and and hence for each n=l,2,..., l/\}( Thus, (Zn(A)} is a bounded sequence whose terms exceed a fixed positive number. 316 Stefan Banach: 4. We now make use of the following theorem (cf. 8. Ba- nach [I, p. 34] and S. Mazur [1]), in which {} and (*!} denote arbitrary bounded sequences of real numbers, a and b denote real numbers, and the symbols lim, lim sup and liminf have their usual meaning: To every bounded sequence {} we can make correspond a num- ber Lim n , termed generalized limit, in such a manner as to fulfil n the following conditions: 1) Lim (ag n +brj n ) = a Lim n + b-Lim ij , /t n n 2 ) lim inf n < Li m $ n < lim sup | n , n n n 5) Jjim . j =^ Jutm f n n The last condition implies that the generalized limit remains unal- tered, when we remove from a sequence a finite number of its terms. Let us now write, for every compact open set A, (4.1) l(A) = Liml n (A). We then have, for any compact open sets A and B: 111 1 . 0<l(A)<+oo; 111 2 . ACS implies l(A)^l(B); iiij. A^B implies l(A)=l(B); iii 5 . Q (A,B)>0 implies l(A + B) = l(A) + l(B). 5. This being so, we denote, for an arbitrary set XCE, by r(X) the lower bound of all the numbers l(A n ) where {A n } is any n ___ sequence of compact open sets such that XC2jA n . We shall show n that the function of a set T, thus defined, fulfils the following con- ditions: 1 We have always 9^F(7) and there exist sets X for which we have 0<r(X)<+oo; this is, in particular, the case of all compact open sets X\ 2 XJ1X % implies 3 XcX n implies 4 e (JCi,Jf 2 )>0 implies 5 Z^Z, implies r(Z 1 )=/ 1 (Z t ). On Haar's measure. 317 Proof. 1. Let X be a compact open set. We have, by defi- nition, F(X)^l(X)<+oa. On the other hand, there clearly exists for each e>0 a (finite or infinite) sequence of compact open sets (A n ) such that X cA n n and F(X) + e^]>jl(A n ). Let 8 be any sphere contained in A'. Since /i the set 8 is closed and compact, this set, and a fortiori the set /S, is already covered by a finite subsequence {A n } of {A n }. In view of iii 2 and Ui 4 , we thus have i Hence, e being arbitrary, it follows that l(S)^F(X) 9 and finally, byiiii, that Q<T(X). 2 and 3 are obvious. 4. Q(XU JT 2 )>0 implies that there exist two open sets G l and G 2 such that X^CG^ X 2 CG 2 and gtG^G^X). On the other hand, there exists for each e >0 a sequence of compact open sets {A n } such that (5.1) Xi + XfEA* and n n Write A ( n } =An-Oi and A^ = A n -O t . Since the sets A? and A are open and compact, and since their distance, like that of G l and G 2 . is positive, we have, on account of iii 6 ' and iii 2 , (5.2) But, since on the other hand X^^A^ and X 2 cA ( 2\ we n n have the inequalities nA^K^^,, 1 ') and r(.Y 2 )<^(A ( n 2) ), so that, n " by (5.2), r(X l ) + r(X 2 )^Sl(^n)- Hence by (5.1), e being arbi- n trary, we obtain r(X l ) + r(X t )^r(X l + X t ) 9 and finally by 3, 5 follows at once from i 3 and iii 3 . 86. It follows from the properties 1 4 of the function r that the latter is an outer measure in the sense of Carathtodory (cf. Chap. Ill, 4) and therefore determines in E a class of sets measurable (Cr), that we shall call, simply, measurable sets. We see at once that for each set X in E the number F(X) is the lower 318 Stefan Banach: bound of the measures (F) of the open sets containing X. It follows, in particular, that Tis a regular outer measure (cf. Chap. II, 6). Finally, since the space E can be covered by a sequence of measurable sets of finite measure (e. g. by a sequence of compact spheres), we easily establish, for the measure F, conditions of meas- urability (fi/ ) similar to those of Theorem 6.6, Chap. III. In part- icular, we shall have: (6.1) In order that a set E be measurable, it is necessary and sufficient that there exist a set ((&<) containing E and differing from it by at most a set of measure zero. 7. We conclude this note by giving two examples of spaces E with the notion of congruence subject to the conditions of 2. Example 1. Let E be a metrical space which is separable and locally compact, and suppose that, among the one to one trans- formations, continuous both ways, by which the whole space E is transformed into the whole space IS, there exists a class dtt of transformations subject to the conditions: 1) Tedll implies T \ $/i; 2) If T l d/l and T 2 f6W, then T^ecVT; 3) For every pair a, b of points of E, there exists a transformation Tfd/l such that T(a)=b; 4) // (a,,) and {b n } are two convergent sequences of points of E such that lim a n = lim b n , and if {T n } is a sequence of transformations n n belonging to cW such that the sequences {^(a,,)} and {T n (b n )} are con- vergent also, then we have lim T n (a n ) = lim T n (b n )- n n Two sets ACE and BCE will be termed congruent, if there exists a transformation TecM? such that T(A) = B (where T(A) denotes the set into which A is transformed, i. e. the set of all the points T(a) for which aeA). It is easy to verify that the conditions ij i 5 are fulfilled. As special cases of such spaces E we may mention: Euclidean ^-dimensional space with cW interpreted as the class of all trans- lations and rotations; the 3-dimensional sphere with dJi interpreted as the class of all rotations. Let us observe that, in the space considered, the sets which are congruent to open sets are themselves open. On the other hand, On Haar's measure. 319 on account of 5, p. 316, the sets congruent to sets of measure (F) zero are themselves of measure zero. It follows therefore from (6.1) that in th3 space E considered the sets which are congruent to meas- urable sets, are themselves measurable. Example 2. Suppose that a metrical space JB, separable and locally compact, constitutes a group, i. e. that with each pair a, ft of elements of E there is associated an element ab of JB, called pro- ductj in such a manner that the following conditions are fulfilled: 1) (ab)c=a(bc) (whatever be the elements a, b and c of E); 2) there exists in E a unit-element 1 such that we have l>a=a-l=a for every acE, 3) to each element aeE there corresponds an inverse element ~ l E which fulfils the equation aa" 1 1. Suppose further that E fulfils the conditions: 4) if lima AI = a and lim b n =bj then ]ima n b n = abj n n n 5) if lima^^a, then lim a^'^a" 1 . n n Given any element ceJB and any set BCE, we denote by cB the set of all the elements aeE such that a=cb where b*B. Given an element a of JB, we write, for every element xe E, T (x) = ax. Thus each element a of E determines a transformation T a , clearly one to one and continuous both ways, of the space E into itself. Denoting the class of all these transformations by eW, we see at once that the conditions 1) 4), p. 318, are fulfilled. In ac- cordance with the definition of congruence employed in Exam- ple 1, two sets A and B in the space in question are congruent if there exists an element c such that B = cA. NOTE II. The Lebesgue integral in abstract spaces by Stefan Banach. Introduction. In this note we intend to establish some general theorems concerning the Lebesgue integral in abstract spaces. This subject has been discussed by several authors (for the references see this volume, pp. 4, 88, 116, 156 and 157). Our considerations differ from those of other writers in that they are not based on the notion of measure. Let us fix a set of arbitrary elements H as an abstract space. We shall denote real functions (i. e. functions which admit real values) defined in H by x(t), y(t), z(t),... where teH, or simply by #,t/,2,... . A set 2 of real functions defined in If will be called linear if any linear combination, with constant coefficients, of two ele- ments of 2, also belongs to 2. Let 2 be a linear set of functions defined in H. A functional F defined in 2 is termed additive if for any pair of elements x and y of 2 and any real number a, we have F(x y) = F(x) + F(y) and F(ax) = a-F(x). The functional F is non-negative if F(x)^0 for any non-negative function x 2. We say that a functional F defined in 2 is a Lebesgue integral (^-integral) in 2 if the following conditions are satisfied: A) The set 2 is linear; B) the functional F is additive and non-negative; C) if 1 (* n }C*i and M f2, 2 \z n (t)\^M(t) for w=l,2,... and and 3 Q limz n (t)^z(t) forte//, then z*H and HmF(z n ) = The Lebesgue integral in abstract spaces. 321 D) if ie 2, -F(z) = and |y(<)|<*(0 for teH, then ye2 and E) if 1 {2 n )C2, n K n +i) for n=l,2,..., 2 lim *(<)=*(<) for teH, and 3lim JF^K+oo, thenzefi and lim J^(^ n ) A * The Lebesgue integrals considered in this note will moreover satisfy the condition: E) If Z 2, then \z\ S. In Part I, a condition is established under which an additive and non-negative functional defined in a linear set of functions (E, may be extended to an T-integral on a certain set 2 containing (. The ^-integral and the set SI will be explicitly defined. In Part II we admit that H is a metrical and compact space. We consider an ^-integral defined in sets containing all functions which are bounded and measurable in the Borel sense. It is shown that each T-integral of this kind is determined by the values which it admits for continuous functions. Conversely, any additive and non- negative functional defined for all continuous functions may be extended as an f-integral to the class of functions measurable (53). We thus obtain the most general ^-integral defined for all functions bounded and measurable (3)). In Part III we deal with an analogous problem supposing that H is the unit sphere of the Hilbert space. In particular, the integral of a continuous function is expressed by explicit formulae. I. Abstract sets. 1. We shall employ the following notations: 1. x^y if x(t)^y(t) for every teH; in particular x^O means that #(0>0 for teJET; 2. |#|=|#(t)| is the modulus of x(t) in the ordinary sense; 3. max(a,y)=H0+y+k y|)> min (^)=i(^+ |-y|); 4. limx n ^=x means that ]imx n (t) = x(t) for ltH\ the relations n n lim sup x n = X) lim inf x n = x are defined similarly; n n 5. x = \(x+\jc\), x={-(x-\x\) (cf. Chap. I, p. 13). 322 Stefan Banach: 2. For the rest of Part I of this note we shall fix a set (E of real functions defined in If, and a functional f(x) defined for xeS, subject to the following conditions: (ij) The set ( is linear; (i 2 ) if xe(, then |a?|e(E; (iij the functional / is additive; (11 2 ) the functional / is non-negative; (11 3 ) if 1 (x n }C<t and M e (, 2 |x n |<Jf for n=l,2,..., and 3 limx n = Q, then limf(x n ) = 0. n n It follows immediately from the conditions (i) that for any pair of elements x and y of (E, max (#,y), min (x,y), x and x also be- long to C. It follows further that the condition (ii 3 ) is equivalent to the following condition: (iii) If 1 (x n }C<i and me(, 2 x n ^m for w=l,2,..., and 3 liminf <>(), then liminf/(j? /1 )^0. 3. We shall establish the following Theorem 1. If the set ( and the functional f satisfy the con- ditions (i) and (ii), then there exists an ^-integral F, defined in a set 2 containing (, such that JF(x) = f(x) whenever #e(; moreover, this integral satisfies the condition E). The proof will result from several lemmas. 4. We denote by S* the set of all functional z(t) defined in H for each of which there exist two sequences {x n }C<, {t/)C(E such that (1) liminf x n ^z^limsupy n . n n It is easily seen that the set 2* is linear and that (CC*. Given a function ze*, we shall term upper ^-integral of z the lower bound of all (finite or infinite) numbers g for each of which there exist a function meg and a sequence of functions {x n } be- longing to G such that x n ^m for n = l,2,..., liminf ,#> 2 and n = liminf /(#). n The definition of the lower ^-integral is analogous to that of the upper T-integral. The upper and lower T-integrals of a function ze 2* will be denoted by p(z) and q(z) respectively. We obviously have q(z)= p( z). The Lebeegue integral in Abstract spaces. 323 5. The sequence !/(#)} in the above definition of the upper ^-integral, may obviously be supposed convergent (to a finite limit or +00). Further, if {#/,)C(, me<&, z^Q, x n ^m for w=l,2,... and ^, then lima? n =0 and consequently, by the condition (ii 3 ), 2, lim/( n )=0. Hence, if zffi*, z>0 and p(z)<P<+oo, n there always exists a. sequence of non-negative functions {x n } belonging to ( such that liminf x n ^z and f(x n )<P for n 1,2,... n Lemma 1. For any function xeti we have p(x)=f(x). Proof. Writing x n x and m x, we have (1) liminf x n ^x and x n ^m for tt=l,2,..., whence p(x)^.f(x). On the other hand, if x l9 x 2j . and m are any functions which belong to (E and satisfy the relations (1), then liminf (x n -x)^Q and x n x^mx for n=l,2,... . It follows from 2, that liminf /(j? n x)^0, i.e. liminf f(x n )^f(jr). Thus fi n , arid finally p(x)^=f(x). Lemma 2. If 2^2*, ^ 2 f 2* an(l tfi moreover, <+, then p( Proof. Let P l and P 2 be arbitrary numbers such that p( and p(z 2 )<P<2- There exist two sequences \x ( n\ {^} of functions belonging to (g and two functions w^ttf and w a *-( such that liminf x^^Zf and lim/(.^ y) )<P y for ; = 1,2 and such that > x n n for )=1,2 and n = l,2,... . Therefore, writing A, jr l) + j?!r we have liminf x^z^z^ and x n ^m for w,=l,2,.... sequently p(^ f 2 a )^:lim/(jr fl )=lim/(.i: ( ll 1) ) f lim/(^ 2) )<A+ P 2> whenc LemmaX. For any function ^^2*, we Proof. Since ?(*)=--- p( 2) (cf. 4), the inequality is obvious if one of the numbers p(z) or p( z) is +co; while, if p(2)<+oo and p(~c:)<4-oo, it follows immediately from Lemma-. 324 Stefan Banach: Lemma 4. If *e2*, p(z)<+oo, then also p(z)<+oo and Proof. Given an arbitrary finite number P>p(z)j there exist a function me(E and a sequence {x n } of functions belonging to ffi such that x n ^m for rc = l,2,..., liminf x n ^z and Urn/ (x n )<P. Note n n that x n ^m, and consequently /(/,)</(#) /(w), for n=l,2,..., whence p|^)<liminf /()< +00. Again and therefore p(z)^p(z) + p(z)j whence, in virtue of Lemma 2, Finally, we mention two propositions which are directly obvious: LemmaS. If z l S,* J 2 2 2* and particular , if t2* and 3^0, then Lemma (i. If 262*, then p(fa)=A'p(z) for any non-negative number X. $6. We shall now denote by 2 the set of all functions z e 2* for which ^(2)^^(^)4=00. The following proposition is an immediate consequence of Lemmas 2 and 6: Lemma 7. If ^ f JJ and z 2 6 2, then (^ 1 + >l 2 z 2 ) e 2 and P(^i^i + ^2^2)^^i?>(-i)+ ^.P( Z 2) for any pair of finite numbers ^ and A 2 . LemmaS. If zefi, ^en ||fi. Proof. Since \z\ z z, it is enough to prove that Ie2 and zeQ. To this end, let us remark that, in virtue of Lemma 4, p()<+, P(?)> and p(z) = p(?) + p(z)j by symmetry, q(z)> co, ^(J)<+oo and ^(5f) = g(^)-f q(z). Since, by hypothesis, p(z) = q(z), it follows that [p(z) q(z)]+[p(z) q(z)] = 0, and so by Lemma 3, p(z) = q(z)= oo and p(z) = q(z) 4= oo. Lemma 9. If z is the limit of a non-decreasing sequence {z n } of functions belonging to 2 and limp ()<+ oo, Mew zt2 and p(z)=]imp(z n ). n Proof. We can clearly assume (by subtracting, if necessary, the function z^ from all functions of the sequence {z n }) that z^O. Writing w n = z n +i&n for n = l,2,..., we shall now follow an argu- ment similar to that of Theorem 12.3, Chap. I. First, we have and p(z n ) = q(z n ) for every w, and so (1) q The Lebesgue integral in abstract spaces. 325 To establish the opposite inequality, let f be an arbitrary positive integer and let us associate (ef. the remark at the begin- ning of 5) with each function w n a sequence {^^1,2, . of non-negative functions belonging to ( such that (2) lim inf x^w n and (3) / Let us write y^--^^. The functions y k clearly belong to ( and, n \ by (2), we have liminf iy^J[V A 2. On the other hand, in virtue A of (3), we find f(y fl )^^p(w ji ) + e^p(z k ^) + e <limp(^) + for n-1 ' A* = 1,2,.... Therefore, p(z) < lim inf f(y.) < lim p(z k ) + e, and A ' A since f is an arbitrary positive number, this combined with (1) gives )-^q(z)-^ lim p(s/?)<+^, which completes the proof. Lemma 1O. If M f 2 and {z,,} t a sequence of junctions belonging Jto C 6't/^ft that |^ n |<Jf /or n^l,2,..., iftfw, putting </=liminf n // - lim sup^ n , ?e ^a^ ^^fi, h e 2, (Consequently, if the sequence {z n } i.s convergent and z lim^, then p (z) = limp(z n ). n Proof. The lemma corresponds to Theoreml2.il, Chap. I, and its proof is analogous to that of the latter. Let us write, for each pair of integers i and j^zi, g..=^ min (z.,2 ^,...,2^). The se- quence {j/ r }._ f . ;+ , is non-increasing, and consequently the sequence {Mgfj}, f . + 1 is non-decreasing. Let sr/^limjf/,. Since the func- tions g. clearly belong to 2, it follows from Lemma 9 that Mg.eZ and p(Mg i ) = limp(Mg u ), i.'e. ^e2 and p(g l )=limp(g lj ). Hence, applying again Lemma 9 to the non-decreasing sequence {g) which converges to 3, we obtain </e2 and By symmetry we have the analogous result for h and the proof is complete. We shall conclude this by mentioning the following lemma which is an immediate consequence of Lemma 5: Lemma 11. If *e2, 2>0 and p(z) = 0, then any function x such that \x\^z belongs to 2 and for any such function x we have p (x) = 0. 326 Stefan Banach: 7. Let F(x) = p(x) for xe. The lemmas of the preceding sections show that the set 2 and the functional F(z) satisfy the theorem stated in 3. Theorem 1 is thus proved. It is easily seen that if an T-integral F^ defined in a linear set S^Dfi satisfies the condition f(x) = F l (x) for #e(E, then F(x) = F l (x) for all xe2. Consequently the functional / determines completely an T-integral in the set 2. II. Metrical compact sets* $8. Let now H be a complete and compact metrical space. We shall specify (E as the set of functions continuous in H. The set g satisfies evidently the conditions (i), 2. It may be shown that any additive and non-negative functional / defined in ( satisfies the condition (ii 3 ) l ). Theorem 1 permits to define a Lebesgue integral F(x) for all functions x belonging to a certain set 8DS , in such a manner that the condition E), p. 321, is satisfied and that F(x) = f(x) for xe (. Evidently, every function x(t) which is constant on H belongs to ffi. It follows by condition C), p. 320, that every bounded function measurable in the sense of Borel belongs to (L We have thus proved the following Theorem 2. Every additive and non-negative functional, defined for att functions which are continuous in a complete compact space H, may by extended to an Z-integral defined in a certain linear set (con- taining all bounded functions measurable in the sense of Borel) so that the condition R) be satisfied. The values of this ^integral for functions bounded and meas- urable (93) are, of course, determined by the given functional /. Hence the most general f-iritegral defined for this class of functions may be obtained by choosing an arbitrary additive non-negative functional defined for all functions which are continuous in H and by extending this functional by means of the method described in Part I of this note. l ) A functional of thin kind is necessarily linear. Every linear functional defined in ( satisfies the condition (ii a ). See S. Banach [I, p. 224]. The Lebesgue integral in abstract spaces. 327 Any linear functional f(x) defined in the set E is the difference of two additive non-negative functionals / 1 (o?) and / 2 (o?) (cf. 8. Ba- nach [I, p. 217]). Extending these functionals by means of Theorem 1 over two sets, 2 l and 2 2 say, respectively, we see that it is possible to extend the functional j(x) over the linear set 2=2! -S 2 . This set will contain all bounded functions measurable ($). The extended ad- ditive functional F(x) evidently satisfies the conditions C) and E), p. 321, and is non-negative. III. The Hilbert space. 9 We shall now understand by H the unit sphere of the Hilbert space, i. e. the set of all sequences {#/} for which . The distance of two points <={#/} and <'={#/} is defined, as usually, by the formula With regard to this definition of distance the space H is not compact and therefore we cannot apply Theorem 2 directly. Let S n be the set of functions x=x(t) = x($u # 2 >---) which are continuous in H and whose values depend only on the first n co- ordinates #/, so that #(#!,#*,. ..) = #(#!, #2,. "AA *---) for an 7 !={*/> IT. Clearly (,,C (+!. 00 It is easily seen that the set (=2?(/i satisfies the conditions n- i (i), 2. Any functional / defined in ( for which the conditions (ii) hold may be extended to an ^-integral defined in a certain set 8 containing (. Lemma 12. The set 2 contain* all bounded functions mesurable ($) defined in H. Proof. Let a? be a bounded continuous function defined in JBT. For any point *H#,,# 2 ,. ..,#,...) and an y positive integer n, we write # A (0~#(iV">#At fV")< Evidently #*< and lima? n =ap. If M is the upper bound of \x(t)\ for ffff, then (j^KJf. Since the constant function z^M certainly belongs to <, it follows from the condition C), p. 320, that o- 328 Stefan Banach: Consequently every bounded and continuous function belongs to fi and by the condition C) the same is true of any bounded func- tion measurable (33). Lemma 13. Every additive and non-negative functional f(x) defined in ( satisfies the condition (ii 3 ), 2. Proof. We define in JET a distance Q^t, t') of two points * = {*i,*2,...}, '=W, *,.-> by We easily verify that with regard to this distance the set // is complete and compact. Let (E be the set of all functions defined in // which are continuous according to the distance defined by the formula (1). Evidently (ECfi. Let / be an additive non-negative functional defined in (. Let x n (t) x(9 l9 ...,#, 0,0, ...) for xe<i and t =(&&...) H. With regard to the distance (1), H is a complete and compact space, and hence the function x(t)e(S, is uniformly continuous. It fol- lows that the sequence {x n } uniformly converges to x. This implies the convergence of the sequence (f(x n )} l ). Let J(x) = lim /(,). If j^O, then x n ^0 for each w, and consequently f(x)^Q. The functional f(x), clearly additive, is therefore non-negative. The set H being compact, it follows, by what has been established in Part TI, that / satisfies in H the condition (ii 3 ) (with (E and / replaced by ( and J respectively). Since (c and T(x) = f(x) for J3e(, the functional / satisfies the condition Hi 3 ) in CB. 10. Now consider an additive non-negative functional f(x) defined in C. Let f n (x) denote the functional defined in ( by the formula (2) f n (x)^f(x) for xetin* We obviously have (3) /(*) = /n+i(*) for xe<tn. l ) Indeed if * > 0, there exists a positive integer N such that e ^x p x (f ^* whenever p>N, q>N. Since the constant function 2=1 belongs to (, we have, for fc=/(l), the inequality kf^f(x p ) -f(s ff )^kr which proves the convergence of <f (*)} The Lebesgue integral in abstract spaces. 329 Conversely, if we choose any sequence {/n(#H of additive non- negative functional, the functional / being defined in ( (where n 1,2,...) subject to the condition (3), then the formula (2) deter- mines an additive non-negative functional f(x) in (. We thus obtain the most general additive non-negative functional f(x) defined in (, and by what has been established in the preceding , the most general Lebesgue integral for all functions bounded and measurable (93). The set ( may be interpreted as the set of all function of n variables #!,...,# which are defined and continuous in the sphere #?++#*<! It is known that the most general additive and non- negative functional defined in & may be represented by a Stjeltjes integral. These general considerations will now be illustrated by the following example. Suppose that the functionals / are given by the formula for Xf G, nJ where <p n denotes a fixed non-negative function integrable in the sphere #?+... -f#^i. The condition (3) may be written in the form To satisfy this condition, we may put, for instance, 9^ 1/2 and ^^\^'~^^ for n^l. We thus obtain (5) rt r 1 \r\...f 1 \r\ ... \r n i Let x be an arbitrary function bounded and continuous in H. We write again x n =x(9i, ...,# n ,0, 0,...). If |#|<M, where M is a constant, then timx n x, 330 Stefan Banach. Now let F be an T-integral which for functions belonging to (E coincides with the functional / subject to (2). 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Condition, of Lipschitz for functions of two variables on an interval, 1 69 ; (1 J ), 192; (17), 192; (l a ), 192; Lusin's, (N), 224; of Lipschitz for functions of a real variable on a set, 264; (T t ), 277; (T,), 277; (S), 282; (D), 290; of Lip- schitz for functions of two variables on a set, 304. Contingent, 263. Coordinates of a point, 56. Cubes, 57; Open, 57; Half open, 57. Covering in the sense of Vitali, 109. Curves, 121; rectifiable on an interval, 122; Rectifiable, 122. Decomposition, Jordan, of a function of a set, 11; Jordan, of a function of an interval, 62; Lebesgue, of a func- tion of a set, 35; Lebesgue, of a func- tion of an interval, 120. Densities, Outer upper and lower, 128; Upper and lower, 129; Right-hand and left-hand, 215. Derivates, General upper arid lower, 106; (Ordinary) upper and lower, 106; Strong upper and lower, 106; Unique, 106; Extreme, 106; Bilateral, 108; Uni- lateral (right- and left-hand) 108; Dini, 108; relative to a set, 108; with res- pect to a function, 108; Intermediate, 108; Symmetrical, upper and lower, 149; with respect to a measure and a sequence of nets in a metrical space, 154; with respect to a sequence of nets of intervals, 188; Approximate right-hand and left-hand, upper and lower, 220; Approximate bilateral, upper and lower, 220; Opposite, 269. Derivative, General, 106; Ordinary, 106; Strong, 106; Unilateral, 108; relative to a set, 108; with respect to a function, 109; with respect to a mea- sure and a sequence of nets in a me- trical space, 154; with respect to a se- quence of nets of intervals, 188; of a complex function, 196; Roussel, 210; Approximate, 220. Diameter, 40. Difference of sets, 5. Differential, Total, 300; Approximate, 300; Upper and lower, 309; Extreme, 309. Direction of a half -line, 262. Distance, of points, 39; of a point and a set, 40; of sets, 40; of points in an Euclidean space, 56; of points In the torus space, 157. Element of a set, 4. Expressions of de Geocze, 171. Faces of an interval, 57 ; of a polyhedron. 164. Families of sets, see Classes of sets. Figure, 58, Elementary, 58. 342 General Index. Function of singularities, of a function of a set, 35; of a function' of an inter- val, 120. Functions (of a complex variable). Com- plex 195; Holomorphic. 195; Deri- vable, 196. Functions (of an interval). 59; Contin- uous, 59; continuous and discontin- uous at a hyperplane, 60; Additive, 61 ; of hounded variation. 61 ; Monoto- ne non -decreasing and non-increasing, 61; Absolutely continuous, 93; Sin- gular, 93; Major and minor, 191; Maj- or and minor, with respect to a func- tion, 207. (See also Functions (of a real variable).) Function* (of a point), Finite6; Character- istic, of sets, 6; Simple, 7; Measurable 12; Integrabie, 20; Upper and lower semi-continuous, 42; Continuous, 42; integrable in the Lebesgue-Stietjes sense, 65; integrable in the Lebesgue sense, 65; Sumrnable, 65; Approx- imately continuous, 132; Kqui-mea- surablc, 143; Cylindrical, 159 (See also Functions (of an interval) and Functions (of a real variable).) Fumtions (of a real triable), 96; of bounded v iriation, 96; Absolutely continuous, 1)6; Singular, 96; Regular, 97; Saltus-, 97; integrable in the sense of Newton, 186; X-integrable, 201; integrable in the sense of Per- ron, 201; Approximately derivable, 220; of bounded variation (in the wide sense) 221; VB, 221; of genera- lized bounded variation (in the wide sense), 221; VIHJ, 221; absolutely continuous (in the wide sense), 223; AC, 223; generalized absolutely con- tinuous (in the \\ide sense), 223; ACG, 223; of hounded variation in the res- tricted sense, 228; VB*, 228; of gene- nilized hounded variation in the res- tricted -en<e, 228; VBG*, 228; abso- lutely continuous in the restricted vense, 231; AC,, 231; generalized ab- solutely continuous in the restricted sense, 231; ACCJ,, 231; ^-integrable, 241; '^-integrable, 241; continuous in the sense of Darboux, 272; Inner ami outer, of a superposition, 286; InvciH', 286. (See also Functions (of an interval) and Functions (of a point).) Functions (of n NT/), Additive, 8; Mono- tone. S; \on decreasing and non-in- crea^inir, s ; Absolutely continuous 3n. f. Singular, 30, 6fi. Functions (of two real variables), of bound- ed variation, 169; Absolutely con tin- uous, 169; Totally differentiate, 300; Differentiable, 300; Approx- imately differentiable, 300. Graph, of a function, 88; of a curve, 121 . Half-plane, 263, 264. Half-space, 263, 306. Half-tangent, Intermediate, 262. Hyperplane, 57; orthogonal to an axis, 57; Intermediate tangent, 263; Ex- treme tangent, 263; Unique tangent, 263; Tangent, 263. Increment of a function, 96. Integrals, Definite, 19, 20, 46, 254; In- definite, 29, 254; Lebesgue-Stieltjes, 65; Lebesgue, 65; Upper and lower Burkill, 166; Definite and indefinite Burkill, 166; Mean value, 178; Perron, 201, y. f 201; Perron-Stieltjes, 207; &-, 207; Den joy, in the wide sen- se, 241; Denjoy-Khintchine, 241; ^-, 241; Denjoy, in the restricted sense, 241; Denjoy-Perron, 241; -*\- f 241; -'-, 241; c<-, 244; Compatible, 254. Interior of a set, 40. Intervals, 57; Closed, 57; Open, 57; Half -open. 57; Degenerate, 57; in the torus space, 157. Length, Outer, of a set, 54; of a set 54; of an interval, 59; of arc of a curve, 122; of a curve, 1-22 Limit, of a sequence of sets, 5; Upper and lower, of a sequence of sets, 5; of a sequence of points, 39; Approx- imate, upper and lower, 218; Approx- imate extreme, 219; Approximate, 219; Approximate bilateral and uni- lateral, 220. Lines, Straight, 56, 57. Maximum, of a function at a point, 42; Strict, 261. Measure, 16; Outer Carathdodory, 43; of a set, 46; Outer, of a set, 46; Reg- ular outer, 50; Outer Lebesgue, 65; Lebesgue, 65; -function, 117; Outer, in the torus space, 157; in the torus space, 158; Outer linear, of a set on a line, 297. Mesh of a net, 153. Minimum, of a function at a point, 42; Strict. 261. General Index. 343 Neighbourhood of a point, 40. Net, of dosed intervals, 57; of half open intervals, 57; in a metrical space, 153; Normal, of intervals, 188. Nucleus of a determining system, 47. Number, Characteristic, of a family of sets, 40. Operation. (A), 47. Ordinate-set of a function, 88. Oscillation, of a function of a point at a point, 42; of a function of an inter- val on a set, 60; of a function of an interval at a set, 60. Parameter, of regularity of a set, 106; of a curve, 121. Part, Common, of sets, 5; Non-negative and non-positive, of a function, 13; Absolutely continuous, of a function of an inteival, 120; Heal and ima- ginary, of a complex function, 195. Plane, V>6, 57; Whole, 264. (See also Hyperplane.) Points, of a space, 6; of accumulation, 40; Isolated, 40; Internal, 40; of a cur- ve. 121; of outer density, 128; of dispersion, 128; of density, 129; Highl- and left-hand, of accumulation, 215; isolated on the right or the left, 215; Unilateral (right- and left-hand),- of outer density, 215; of linear density in the direction of an axis, 298. Polyhedron, 164. Portion of a set, 41. Primitive of Neirton, 186. Product of sets, 5; Combinatory, 82; ('ar- tesian, 82. Radius, of a neighbourhood, 40; of an open sphere, 40; of a closed sphere, 40. Relations, Denjoy, 269. Sequences, Convergent, of sets, 5; As- cending and descending, of sets, 5; Non-decre'ising and non-increasing, of sets, 5; Monotone, of sets, 5; Con- vergent, of points, 40; Regular, of nets of intervals, 57 ; Regular, of nets in a metrical space, 153; of mean value integrals, 178; Normal, of nets of intervals, 188; Binary, of nets of intervals, 191. Sets, Empty, 4; Enumerable, 4; Meas- urable, 7; Bounded, 40; Derived, 40; Closed, 40; Isolated, 40; Perfect, 40; Open, 40; Non -overlapping, 40; (J), 40; ((&), 40; (*), 41; measurable ($), 4 1 ; Borel, 4 1 ; closed in a set, 4 1 ; open in a set, 41 ; everywhere dense in a set, 41; non-dense in a set, 41; of the, first and the second category (in a bet), 41 ; Separable, 41; measurable with res- pect to an outer Caratheodory meas- ure, 14; regular with respect to an outer Caratheodory measure, 50; of zero length, 54; of zero .irea, 54; of zero volume, 54; of finite length, 54; of finite area, 54; of finite volume, 54; Linear, 56; Plane, 56; Ordinate-, 88; Measurable, in the torus fpace, 158; Cylindrical, in the torus space, 159. Side of a hyperplane, 263; Empty, 263. Space, Abstract, 6; Metrical, 39; Com- plete, 54; Euclidean, 56; Torus, 157; Whole, 306. Sphere, Open, 40; Closed, 40. Square, 57. Subdivision of a figure. 165. Subsets, 4. Sum of sets, 5. Superposition of functions, 286. Surfaces, Continuous, 164. System, Determining, 47; Degenerate determining, 49. Tangent, Intermediate, 263; Extreme, 263; Unique, 263. Torus space, 157. Translation, of a set by a vector, 91; in the torus apace, 158. Variation, Relative (upper and lower) and absolute, of a function of a set, 10; Relative (upper and lower) and ab- solute, of a function of an interval, 61, 62; of a function of a real variable, 96; of a function of an interval at a set, 166; Weak, 221; Strong, 228. Vertex, of a polyhedron, 164; of an angle, 310. Volume , of an interval, 59; of an interval in the torus space, 157. NOTATIONS. CHAPTERI. e,4; C<M254)O, = ,4; -2,5(258); /T,-f , ,liminf, lirasup,5; lim, 5 (39); ((), d 69 $ (where is a class of sets), 5; C, 6; C E (where E is a set), 6; <!>,,!,; ...; v n , E n } (where E i9 -..,E n are sets and v,,...,v n numbers), 7; W, J,W, 10(61,96); E, 12; /, ^ (where / is a function), 13; ($){/o> (where is a class of sets, / a function of a point and <M a measure), 20. CHAPTER II. (>, 39 (40); lim, 39 (5) ;<>,<*, 4,40; 8,41(122,165); 8,41;^', A, A (where A is a set), 42; (5, ft, 40; 93, 41; M, m, 42; o, 42 (60); 2 r (where Pi& an outer Carath^odory measure), 44; (r),j*/aT (where/ 1 is an outer Carath^odory measure and / a function of a point), 46; N,47; A, 53. CHAPTER III. 2*, 56; (x l9 ... 9 x m ) (where x lt ...,x m are real numbers), 56; [a 1 ,6 1 ;-...;a /n ,6 m ], [a,, b } ; ...; a m , b m ) 9 (a,, 6,; ...;a m , 6J, (a,, 6,, ...; a m , 6J (where Opfc,, ...,a m , & m are real numbers), 57; 0, , 59; 0,60(254); o, 60(42); W, W,W, 61 (10, 96); U* (where U is a function of an interval), 64; $jdU (where/is a function of a point and U a function of an interval), 64; L, 1 1, 65 (117); f(x)dx (where / is a function of a point), 66; x , 76 (82); W (where U and F are functions of an interval), 76; x,82(76); 9E3) (where 9E and 9) are classes of sets), 82; f*v (where p and v are measures), 86; 1* (where $ is a class of sets and rja measure), 87; A, 88; B, 88 (121); Q (a} (where Q is a set and a a point), 91; W, W, W, 96 (10,61); /(a-f), /(a ) (where / is a function arid a a point), 97; F(I) (where F is a function of a real variable and 7 an interval), 99; F[E] (whore F is a func- tion of a real variable and E a set), 100. CHAPTER IV. ^0,0, F E ,F E >F E , F'^F^F^J, F~jj, F'y (where 12 isa set, and F, U functions of a real vari- able), 108, 109; L, 117(65); p, 121; B, 121 (88); S, 122 (41, 165);/ a ,f \f\f (where / is a function of a real variable), 144; f (where / is a function of two real vari- ables),147;D sym ,P sym , 149; (p, 9R) 1), (f* 9 SW) D, (/i, 2W) D (where ^ is a measure and 3W a sequence of nets), 154 ( 106); Q w , 157; Q m , ' m9 s'^ (where m is an integer and f a point in the torus space), 157; L w , L*, 157; _j- 158. CHAPTER V. S S 165 (41, 122); [7(3) (where U is a function of an interval and 3 a system of intervals), 165; Jlf, |[7, Jt7 (where J7 is a function of an inter- val), 166; W lf W t . 169; Gt.G^G, 171;"^, H 2 , H, 174. CHAPTER VI. (SR).F, ($l)F, (W)F' (where W is a sequence of nets of inter- vals and F a function of an interval), 188; J lf J 2 , J, 195; (^)j*, 201; (#$)]*, 208. CHAPTER VII. (~oo, a), (00, a], (a,-f oo), fa. -f oo) (where a is a real number), 215; lim sup ap, lim inf ap, lim ap, 219; F+, F^ p , ^ ap , F ap , ^ ap ,^ ap , ^ p (where .F is a function), 220; V,221; V, 228. CHAPTER VIII. (5>)|, ($>,)]*, 241; (5)| f 244; 0,254(60); C 254 (4); C , (T H , S H * (where is an integral), 255 and 256; 2, 258 (5). CHAPTER IX. contg, 263; F^F^^F^F^ ^ F~^ ,.F' (where JP is a function of two variables), 298; liminf^, limsup^, lim^; (where E is a set), 304. CONTENTS. pages PREFACE Ill CHAPTER I. The integral in an abstract spaee. 1. Introduction 1 2. Terminology and notation 4 3. Abstract space X 6 4. Additive classes of sets 7 5. Additive functions of a set 8 6. The variations of an additive function 10 7. Measurable functions 12 8. Elementary operations on measurable functions 14 9. Measure 16 10. Integral 19 11. Fundamental properties of the integral 21 12. Integration of sequences of functions 26 13. Absolutely continuous additive functions of a set 30 14. The Lebesgue decomposition of an additive function 32 15. Change of measure 36 CHAPTER II. Carat h^odory measure. 1. Preliminary remarks 39 2. Metrical space 39 3. Continuous and semi-continuous functions 42 4. Carathe"odory measure 43 5. The operation (A) 47 6. Regular sets 50 7. Borel sets 51 8. Length of a set 53 1 9. Complete space 54 CHAPTER III. Functions of bounded variation and the Lebesgue- Stieltjes integral. 1. Euclidean spaces 56 2. Intervals and figures 57 3. Functions of an interval 59 4. Functions of an interval that are additive arid of bounded variation 61 5. Lebesgue-Stieltjes integral. Lebesgue integral and measure .... 64 6. Measure defined by a non -negative additive function of an interval 67 7. Theorems of Lusin and Vitali-Caratheodory 72 8. Theorem of Fubini 76 9. Fubini's theorem in abstract spaces 82 10. Geometrical definition of the Lebesgue-Stieltjes integral .... 88 11. Translations of sets 90 12. Absolutely continuous functions of an interval 93 13. Functions of a real variable 96 14. Integration by parts 102 346 CHAPTER IV. Derivation of additive functions of a set and of an interval. 1. Introduction 105 2. Derivates of functions of a set and of an interval 106 3. Vitali's Covering Theorem 109 4. Theorems on rneasurability of derivates 112 5. Lebesgue's Theorem 114 6. Derivation of the indefinite integral 117 7. The Lebesgue decomposition 118 8. Rectifiable curves 121 9. De la Valise Poussin's theorem 126 10. Points of density for a set 128 11. Ward's theorems on derivation of additive functions of an interval 133 12. A theorem of Hardy -Littlewood 142 13. Strong derivation of the indefinite integral 147 14. Symmetrical derivates 149 15. Derivation in abstract spaces 152 16. Torus space 157 CHAPTER V. Area of a surface z^Ffay). 1. Preliminary remarks 163 2. Area of a surface 164 3. The BurkOl integral 165 4. Bounded variation and absolute continuity for functions of two variables 1 69 5. The expressions of de Geocze . . . . 171 6. Integrals of the expressions of de Geocze 174 7. Rad6's Theorem 177 8. Tonelli's Theorem - 181 CHAPTER VI. Major and minor functions. 1. Introduction 186 2. Derivation with respect to normal sequences of nets 188 3. Major and minor functions . . . , 190 4. Derivation with respect to binary sequences of nets 191 6. Applications to functions of a complex variable 195 6. The Perron integral 201 7. Derivates of functions of a real variable 203 8. The Perron-Stieltjes integral 207 CHAPTER VII. Functions of generalized bounded variation. 1. Introduction 213 2. A theorem of Lusin 215 3. Approximate limits and derivatives 218 | 4. Functions VB and VBG 221 5. Functions AC and ACG 223 6. Lusin's condition (N) 224 7. Functions VB, and VBG, 228 8. Functions AC, and ACG, 231 9. Definitions of Denjoy-Lusin 233 10. Criteria for the classes of functions VBG,, ACG,. VBG and ACG 234 347 CHAPTER VIII. Denjoy integrals. 1. Descriptive definition of the Denjoy integrals 241 2. Integration by parts 244 3. Theorem of Hake-Alexandroff-Looman 247 4. General notion of integral 254 5. Constructive definition of the Denjoy integrals 256 CHAPTER IX. Derivates of functions of one or two reai variables. 1. Some elementary theorems 260 2. Contingent of a set 262 3. Fundamental theorems on the contingents of plane seta .... 264 4. Denjoy'a theorems 269 5. Relative derivates 272 6. The Banach conditions (T t ) and (T t ) 277 7. Three theorems of Banach 282 8. Superpositions of absolutely continuous functions 286 9. The condition (D) 290 10. A theorem of Denjoy-Khintchine on approximate derivates . . . 295 | II. Approximate partial derivates of functions of two variables . . . 297 $ 12. Total and approximate differentials 300 13. Fundamental theorems on the contingent of a set in space . . . 304 14. Extreme differentials 309 NOTE I by S. BANACH. On Haar's measure 314 NOTE II by S. BANACH. The Lebesgue integral in abstract spaces 320 BIBLIOGRAPHY 331 GENERAL INDEX 341 NOTATIONS 344 ERRATA The first numbers refer to pages, the denote lines counted for: 4,14* P. J. Daniell [2]. 4,12* S. Bochner [1], 6,14* a/0=+oo. 17,9* W. Sierpinski [3], 30.13* P. J. Daniell [2] 44,13* spaces. 44,1719 93,9 110,3 163.3* all the sets X for which F(X)=0 (in particular, it includes the empty set). authors. T. Rad6 [I;1; second to lines; starred numbers from the foot. read: P. J. Daniell [1;4;6], G.Birkhoff[l], S. Bochner [1], N. Dunford [2;3], o/O ^=00 according as a>0 or a- 0. W. Sierpinski [14], P. J. Daniell [4] spaces. (For various examples of Carath^odory measures in metrical spaces see also A. Haar [1] and A. Appert [1].) the empty set aivd all the sets X for which authors. Cf. also N. Dunford [11. + l) m -u 1 . a VA 1 n |-- > (4w + l) BI -a~' 1 .|af| n T. Rad6[I;l;4j; also E.J. McShane [Ij, C. B. Morrey[l] and T. Rado [5]